Peter’s name was Simon.
Jesus gave him the new name, Peter, or Petrus. It was similar to God renaming Abram as Abraham in the O.T., and Jacob as Israel, as a sign that they were chosen.
Petrus also means rock. So the new name enabled the play on words that Jesus used when addressing Peter.
Yes, in another verse, Jesus calls himself the Cornerstone. And he is also speaking about himself in the parable about the house built on sand as opposed to the house built on a firm foundation.
But in this particular verse, he is clearly talking about Peter as the Rock on which He will build His Church.
This reading goes back to the earliest days of the Church, and the writings of the Church Fathers.
Now I know you are Cicero, and upon this rock I build my faith.