The Catechism of the Catholic Church contains a section on the line of the Apostles' Creed reading (in translation) "and in Jesus Christ, His only Son, our Lord". Each phrase is discussed separately; but unfortunately the five paragraphs discussing the phrase "His only Son" focus more on "Son" than on "only".
St. Thomas Aquinas, however, touches on the issue in the Summa Theologica. The First Part of this work contains a large number of "questions" (the Latin might be better translated "discussions" or "investigations" regarding the Trinity. In particular, Question 27, Article 5 (and again the Latin might be better translated "point" or "section") asks "Whether there are more than two processions in God?"
Aquinas explains the relationship of the Father to the Son by offering the analogy of a person who has in mind something they want to say. In one sense, the word they want to say is distinct from them; they are not identical to what they say. In another sense, though, the word (especially before it is spoken) is a part of them: it is a thought in their mind. Thus, Aquinas says, the Son is to the Father, proceeding from Him as one's word proceeds from one's mind and yet part of Him as one's words and thoughts are part of who one is. (In a similar way, the Spirit proceeds from the Father and the Son as a person's love proceeds from them and from their words.)
Aquinas derives from this analogy an objection to God's having only one Son: "In us there is not only one procession of the word, but there are many: for in us from one word proceeds another. ... Therefore in God there are more than two processions." (Question 27, Article 5, Objection 3) That is, he objects that since we can have multiple words proceeding from us, surely God could have multiple Words (that is, multiple Sons of God) proceeding from Him?
His response is to say that God is not really like us in this sense. Words, in us, proceed from our understanding of a situation; and our need to use multiple words arises from the fact that our understanding comes in parts and stages. We don't automatically understand everything perfectly all at once, nor can we (therefore) communicate everything perfectly all at once. But God can. He has therefore, Aquinas says, only one Word by which he understands everything, and communicates everything: "God understands all things by one simple act. ... Hence there cannot exist in Him a procession of Word from Word ... for there is in Him only one perfect Word." (Reply to Objection 3)
This unique Word of God we call the Son of God (as Aquinas discusses in Question 27 Article 2); and thus there can be only one Son of God.