How has Roman Catholic Confession changed throughout the centuries?
I know that this question deals with the 16th century on, but I would like to give a little background first.
According to the Council of Trent, the consensus of all the Fathers always understood that by the words of Christ just cited, the power of forgiving and retaining sins was communicated to the Apostles and their lawful successors (Sess. XIV, c. i). It is therefore Catholic doctrine that the Church from the earliest times believed in the power to forgive sins as granted by Christ to the Apostles. Such a belief in fact was clearly inculcated by the words with which Christ granted the power, and it would have been inexplicable to the early Christians if any one who professed faith in Christ had questioned the existence of that power in the Church. But if, contrariwise, we suppose that no such belief existed from the beginning, we encounter a still greater difficulty: the first mention of that power would have been regarded as an innovation both needless and intolerable; it would have shown little practical wisdom on the part of those who were endeavouring to draw men to Christ; and it would have raised a protest or led to a schism which would certainly have gone on record as plainly at least as did early divisions on matters of less importance. But no such record is found; even those who sought to limit the power itself presupposed its existence, and their very attempt at limitation put them in opposition to the prevalent Catholic belief.
Turning now to evidence of a positive sort, we have to note that the statements of any Father or orthodox ecclesiastical writer regarding penance present not merely his own personal view, but the commonly accepted belief; and furthermore that the belief which they record was no novelty at the time, but was the traditional doctrine handed down by the regular teaching of the Church and embodied in her practice. In other words, each witness speaks for a past that reaches back to the beginning, even when he does not expressly appeal to tradition.
St. Augustine (d. 430) warns the faithful: "Let us not listen to those who deny that the Church of God has power to forgive all sins" (De agon. Christ., iii).
St. Ambrose (d. 397) rebukes the Novatianists who "professed to show reverence for the Lord by reserving to Him alone the power of forgiving sins. Greater wrong could not be done than what they do in seeking to rescind His commands and fling back the office He bestowed. . . . The Church obeys Him in both respects, by binding sin and by loosing it; for the Lord willed that for both the power should be equal" (On Penance I.2.6).
Again he teaches that this power was to be a function of the priesthood. "It seemed impossible that sins should be forgiven through penance; Christ granted this (power) to the Apostles and from the Apostles it has been transmitted to the office of priests" (On Penance II.2.12).
The power to forgive extends to all sins: "God makes no distinction; He promised mercy to all and to His priests He granted the authority to pardon without any exception" (On Penance I.3.10).
Against the same heretics St. Pacian, Bishop of Barcelona (d. 390), wrote to Sympronianus, one of their leaders: "This (forgiving sins), you say, only God can do. Quite true: but what He does through His priests is the doing of His own power" (Ep. I ad Sympron., 6 in P.L., XIII, 1057).
In the East during the same period we have the testimony of St. Cyril of Alexandria (d. 447): "Men filled with the spirit of God (i.e. priests) forgive sins in two ways, either by admitting to baptism those who are worthy or by pardoning the penitent children of the Church" (In Joan., 1, 12 in P.G., LXXIV, 722).
St. John Chrysostom (d. 407) after declaring that neither angels nor archangels have received such power, and after showing that earthly rulers can bind only the bodies of men, declares that the priest's power of forgiving sins "penetrates to the soul and reaches up to heaven". Wherefore, he concludes, "it were manifest folly to condemn so great a power without which we can neither obtain heaven nor come to the fulfillment of the promises. . . . Not only when they (the priests) regenerate us (baptism), but also after our new birth, they can forgive us our sins" (On the Priesthood III.5 sq.).
St. Athanasius (d. 373): "As the man whom the priest baptizes is enlightened by the grace of the Holy Ghost, so does he who in penance confesses his sins, receive through the priest forgiveness in virtue of the grace of Christ" (Frag. contra Novat. in P.G., XXVI, 1315).
These extracts show that the Fathers recognized in penance a power and a utility quite distinct from that of baptism. Repeatedly they compare in figurative language the two means of obtaining pardon; or regarding baptism as spiritual birth, they describe penance as the remedy for the ills of the soul contracted after that birth. But a more important fact is that both in the West and in the East, the Fathers constantly appeal to the words of Christ and given them the same interpretation that was given eleven centuries later by the Council of Trent. In this respect they simply echoed the teachings of the earlier Fathers who had defended Catholic doctrine against the heretics of the third and second centuries. Thus St. Cyprian in his "De lapsis" (A.D. 251) rebukes those who had fallen away in time of persecution, but he also exhorts them to penance: "Let each confess his sin while he is still in this world, while his confession can be received, while satisfaction and the forgiveness granted by the priests is acceptable to God" (c. xxix). (See LAPSI.) The heretic Novatian, on the contrary, asserted that "it is unlawful to admit apostates to the communion of the Church; their forgiveness must be left with God who alone can grant it" (Socrates, Church History V.28). Novatian and his party did not at first deny the power of the Church to absolve from sin; they affirmed that apostasy placed the sinner beyond the reach of that power — an error which was condemned by a synod at Rome in 251. - Catholic Encyclopedia
In the Early Church, confessions was done publicly.
Three kinds of penance are to be distinguished canonical, prescribed by councils or bishops in the form of "canons" for graver offences. This might be either private, i.e., performed secretly or public i.e., performed in the presence of bishop, clergy and people. When accompanied by certain rites as prescribed in the Canons, it was solemn penance.
Public penance did not necessarily include a public avowal of sin. As St. Augustine also declares, "If his sin is not only grievous in itself, but involves scandal given to others, and if the bishop [antistes] judges that it will be useful to the Church [to have the sin published], let not the sinner refuse to do penance in the sight of many or even of the people at large, let him not resist, nor through shame add to his mortal wound a greater evil" (Sermo cli, n. 3). - Catholic Encyclopedia
It was the the Church in Ireland in particular followed by the Churches in the British Isles that the notion and practice of private confession and private penance first became the the norm that the Catholic Church took up as the preferred way to administer the sacrament of confession.
In the British and Irish Churches
The penitential system in these countries was established simultaneously with the introduction of Christianity, was rapidly developed by episcopal decrees and synodal enactments, and was reduced to definite form in the Penitentials. These books exerted such an influence on the practice in Continental Europe that, according to one opinion, they "first brought order and unity into ecclesiastical discipline in these matters" (Wasserschleben, "Bussordnungen d. abendlandischen Kirche", Halle, 1851, p. 4. — For a different view see Schmitz, "Die Bussbucher u. die Bussdisciplin d. Kirche", Mainz, 1888, p. 187). In any case, it is beyond question that in their belief and practice the Churches of Ireland, England, and Scotland were at one with Rome. The so-called Synod of St. Patrick decrees that a Christian who commits any of the capital sins shall perform a year's penance for each offence and at the end shall "come with witnesses and be absolved by the priest" (Wilkins, "Concilia", I, p. 3). Another synod of St. Patrick ordains that "the Abbot shall decide to whom the power of binding and loosing be committed, but forgiveness is more in keeping with the examples of Scripture; let penance be short, with weeping and lamentation, and a mournful garb, rather than long and tempered with relaxations "(Wilkins, ibid., p. 4). For various opinions regarding the date and origin of the synods, see Haddan and Stubbs, "Councils", II, 331; Bury, "Life of St. Patrick", London, 1905. The confessor was called anmchara (animae carus), i.e., "soul's friend". St. Columba was anmchara to Aidan, Lord of Dalraida, A.D. 574 (Adamnan's "Life of St. Columba", ed. Reeves, p. lxxvi); and Adamnan was "soul's friend" to Finnsnechta, Monarch of Ireland, A.D. 675 (ibid., p. xliii). The "Life of St. Columba" relates the coming of Feachnaus to Iona, where, with weeping and lamentation, he fell at Columba's feet and "before all who were present confessed his sins. Then the Saint weeping with him, said to him: 'Arise, my son and be comforted; thy sins which thou hast committed are forgiven; because, as it is written, a contrite and humble heart God doth not despise,'" (ibid., I, 30). The need and effects of confession are explained in the Leabhar Breac: "Penance frees from all the sins committed after baptism. Every one desirous of a cure for his soul and happiness with the Lord must make an humble and sorrowful confession; and the confession with the prayers of the Church are as baptisms to him. As sickness injures the body, so sin injures the soul; and as there is a cure for the disease of the body, so there is balm for that of the soul. And as the wounds of the body are shown to a physician, so, too, the sores of the soul must be exposed. As he who takes poison is saved by a vomit, so, too, the soul is healed by confession and declaration of his sins with sorrow, and by the prayers of the Church, and a determination henceforth to observe the laws of the Church of God. . . Because Christ left to His Apostles and Church, to the end of the world, the power of loosing and binding."
Slowly but surely the the Church in the medieval period developed this wonderful sacrament into the shape and form we see today.
That said the administration of this sacrament since the 16th century has seen very little changes in the form of administration of this sacrament of reconciliation.
In the question mentioned above the OP mentions that it is the common practice to go to whatever confessor (priest) that is present at a Catholic school for confession.
Just because this is the most common way to go to confession, that does not mean that that this is an absolute norm.
A priest who has the faculties to hear confessions can also carry the title of confessor, especially if the penitent sees a particular priest on a very regular bases for this sacrament.
Religious more often than not have a unique confessor to confess their sins to. Having a single priest to hear one’s sins has more benefits than going to just a random priest. A true confessor in this case can help the faithful (most often a Religious as laypeople generally go to their local pastor or chaplain) advance in holiness, avoiding the occasions of temptations and much more as the priest in this case the know the soul of the penitent much more intimately.
The practice is not limited to Religious and I would highly recommend this to everyone.
It is also true that he confessors of Religious are generally appointed to a particular convent or Religious Order, it is not an absolute rule or guideline. Generally speaking Benedictine Religious (monks or nuns) will have a number of spiritual confessors that are also Benedictine priests also typically called a spiritual director.
This is the type of confessor that St. Theresa of the Child Jesus and St. Gemma had. This option has always been available to go to confession to a priest in whom one is more comfortable with and in whom one is at ease in gaining more spiritual insights on how to become a saint.
Another manner to confess one’s sins is called as St. Francis de Sales and other saints is a general confession.
A General Confession as understood by St. Ignatius of Loyola is a form of Confession whereby one spends 3 to 10 days preparing for a confession of all one's 'sins up to that time.' The main goal of the "general confession" is to turn one's life from one of sin to a more devout one. The Spiritual Exercises of St. Ignatius have done much to popularise this form of confession, with such a confession being the significant end-point of the First Week of his Spiritual Exercises.
Although not necessary for eternal salvation, this is the practice of many Religious prior to taking simple vows.
In closing I would like to simply add one more point. And this is new (1986). This sacrament is so important that in the Code of Canon Law, the Church allows the faithful to go to confession to non-Catholic priests with valid sacraments such as in the Orthodox Churches in the case of an emergency.
Can. 844 §1. Catholic ministers administer the sacraments licitly to Catholic members of the Christian faithful alone, who likewise receive them licitly from Catholic ministers alone, without prejudice to the prescripts of §§2, 3, and 4 of this canon, and can. 861, §2.
§2. Whenever necessity requires it or true spiritual advantage suggests it, and provided that danger of error or of indifferentism is avoided, the Christian faithful for whom it is physically or morally impossible to approach a Catholic minister are permitted to receive the sacraments of penance, Eucharist, and anointing of the sick from non-Catholic ministers in whose Churches these sacraments are valid.
Thus since the 16th century very little has truly changed in form of this sacrament. We should simple be aware that other forms in the administration of this wonderful sacrament have always traditionally existed.