I would always begin any inquiry about a particular quotation by looking at it within its original context and then by comparing it to other writing of the same author. I have encountered and studied a similar quotation by St. Augustine to yours that I will share below.
**“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 also built. For other foundation
no man lay that this is laid, which is Christ Jesus.” **
from Augustine, Homilies on the Gospel of John, Tractate 124; John 21:19-25
Like the quotation you offer, taken out of context, this particular quotation seems to be refuting the doctrine of the Papacy and the primacy of Peter as “the Rock.”
However, in any of his writings, taken within context, St. Augustine does not at all question or refute the primacy of Peter. And no where in any of this writing does he deny the authority of the pope. In fact, just three sentences before the quotation I have cited, he states: “So does the Church act in the blessed hope through this troublous life; 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.” (Augustine, Homilies on the Gospel of John, Tractate 124; John 21:19-25).
Here, St. Augustine is emphasizing his point about Peter’s role being a mere portion of the power and authority of Christ. He prefaces this quotation with: “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.” He goes on to expound on the inferior nature of Peter to that of Christ. He does not in any way deny Peter’s authority or primacy in the Church, but rather reinforces it with the proper perspective of Jesus as the original source and summit of “the rock.”
If you haven’t read the whole homily (Tractate 124), I would just so encourage you to do so! It is so beautiful! It is truly amazing how deeply profound and insightful St. Augustine’s spirituality is. He explains John 21: 19-25 in such a profitable way. Lots of goosebump moments. He really draws the connection between St. Peter and Christ and shows how St. Peter’s mission was truly one in the footsteps of the Lord. And how our suffering and trials of this life truly connect us with Christ, too.
Likewise, I would encourage you to read any quotation within its original context and have some other writings of the same author to compare and contrast the spirit and the letter!