I was under the impression that a "vicar" was a "replacement" or "one acting in the stead of" another (an "agent") but it appears the Roman title of "vicar" seems to be more of an assistant. What is the intended significance of the title as applied to Catholic Popes? Was their title part of Roman government or just a religious title?
Was the Catholic title “vicar of Christ” conceived as an “assistant” or an “agent”?
The short answer is: The original theological connotations of the title had a pastoral sense to it, evoking the words of Christ to the Apostle Peter, regarded by the first Catholic Pope in John 21:16-17, "Feed my lambs... Feed my sheep", so Christ made Peter his vicar.
The Catholic Encyclopedia explains the title of the Vicar of Christ as follows:
A title of the pope implying his supreme and universal primacy, both of honour and of jurisdiction, over the Church of Christ. It is founded on the words of the Divine Shepherd to St. Peter: "Feed my lambs. . . . Feed my sheep" (John 21:16-17), by which He constituted the Prince of the Apostles guardian of His entire flock in His own place, thus making him His Vicar and fulfilling the promise made in Matthew 16:18-19.
In the course of the ages other vicarial designations have been used for the pope, as Vicar of St. Peter and even Vicar of the Apostolic See (Pope Gelasius, I, Ep. vi), but the title Vicar of Christ is more expressive of his supreme headship of the Church on earth, which he bears in virtue of the commission of Christ and with vicarial power derived from Him. Thus, Innocent III appeals for his power to remove bishops to the fact that he is Vicar of Christ (cap. "Inter corporalia", 2, "De trans. ep."). He also declares that Christ has given such power only to His Vicar Peter and his successors (cap. "Quanto", 3, ibid.), and states that it is the Roman Pontiff who is "the successor of Peter and the Vicar of Jesus Christ" (cap. "Licet", 4, ibid.). The title Vicar of God used for the pope by Nicholas III (c. "Fundamenta ejus", 17, "De elect.", in 6) is employed as an equivalent for Vicar of Christ.
Wikipedia adds the follows:
Vicar of Christ (from Latin Vicarius Christi) is a term used in different ways and with different theological connotations throughout history. The original notion of a vicar is as an "earthly representative of Christ", but it's also used in the sense of "person acting as parish priest in place of a real person." The title is now used in Catholicism to refer to the bishops and more specifically to the Bishop of Rome (the pope).
The use of the term Vicar of Christ appears in the 5th century, in a synod of bishops to refer to Pope Gelasius I. The theological connotations of the title got a pastoral sense, evoking the words of Christ to the Apostle Peter, regarded by the first Catholic Pope in John 21:16-17, "Feed my lambs... Feed my sheep", so Christ made Peter his vicar and pastor with the responsibility to feed his flock (i.e. the Church) in his own place.