Augustine
"Among these [apostles] Peter alone almost everywhere deserved to represent the whole Church. Because of that representation of the Church, which only he bore, he deserved to hear ‘I will give to you the keys of the kingdom of heaven’" (Sermons 295:2 [A.D. 411]).
"Some things are said which seem to relate especially to the apostle Peter, and yet are not clear in their meaning unless referred to the Church, which he is acknowledged to have represented in a figure on account of the primacy which he bore among the disciples. Such is ‘I will give unto you the keys of the kingdom of heaven,’ and other similar passages. In the same way, Judas represents those Jews who were Christ’s enemies" (Commentary on Psalm 108 1 [A.D. 415]).
"Who is ignorant that the first of the apostles is the most blessed Peter?" (Commentary on John 56:1 [A.D. 416]).
While all the apostles were given the power to bind and loose, only Peter was given the keys; it's not an incidental thing.
"Some things are said which seem to relate especially to the apostle Peter, and yet are not clear in their meaning unless referred to the Church, which he is acknowledged to have represented in a figure on account of the primacy which he bore among the disciples. Such is ‘I will give unto you the keys of the kingdom of heaven,’ and other similar passages. In the same way, Judas represents those Jews who were Christ’s enemies" (Commentary on Psalm 108 1 [A.D. 415]).
"In the same way, Judas represents those Jews who were Christ’s enemies":
... One wicked man represents the whole body of the wicked; in the same way as Peter, the whole body of the good, yea, the body of the Church, but in respect to the good. For if in Peter's case there were no sacramental symbol of the Church, the Lord would not have said to him, 'I will give unto thee the keys of the kingdom of heaven: whatsoever thou shalt loose on earth shall be loosed in heaven; and whatsoever thou shalt bind on earth shall be bound in heaven.' If this was said only to Peter, it gives no ground of action to the Church. But if such is the case also in the Church, that what is bound on earth is bound in heaven, and what is loosed on earth is loosed in heaven, for when the Church excommunicates, the excommunicated person is bound in heaven; when one is reconciled by the Church, the person so reconciled is loosed in heaven: if, such, then is the case in the Church, Peter, in receiving the keys, represented the good in the Church, and in Judas' person were represented the bad in the Church...
Augustine
"Who is ignorant that the first of the apostles is the most blessed Peter?"
(Commentary on John 56:1 [A.D. 416]).
From the same commentary:
And this Church, symbolized in its generality, was personified in the Apostle Peter, on account of the primacy of his apostleship. For, as regards his proper personality, he was by nature one man, by grace one Christian, by still more abounding grace one, and yet also, the first apostle; but when it was said to him, 'I will give unto thee the keys of the kingdom of heaven, and whatsoever thou shalt bind on earth, shall be bound in heaven; and whatsoever thou shalt loose on earth, shall be loosed in heaven,' he represented the universal Church, which in this world is shaken by divers temptations, that come upon it like torrents of rain, floods and tempests, and falleth not, because it is founded upon a rock (petra), from which Peter received his name. For petra (rock) is not derived from Peter, but Peter from petra; just as Christ is not called so from the Christian, but the Christian from Christ. For on this very account the Lord said, 'On this rock will I build my Church,' because Peter had said, 'Thou art the Christ, the Son of the living God.' On this rock, therefore, He said, which thou hast confessed, I will build my Church. For the Rock (Petra) was Christ; and on this foundation was Peter himself built. For other foundation can no man lay than that is laid, which is Christ Jesus. The Church, therefore, which is founded in Christ received from Him the keys of the kingdom of heaven in the person of Peter, that is to say, the power of binding and loosing sins. For what the Church is essentially in Christ, such representatively is Peter in the rock (petra); and in this representation Christ is to be understood as the Rock, Peter as the Church.
St. Augustine, On the Gospel of John, Tractate 124.5.
Cordially,