He is Roman, because he is Italian, and also because the Holy See is now in Rome.
The Peter part I have yet to understand, but, Malachy's other mottoes were not only similarly opaque, some were a pretty good reach.
Maybe the Rock refers to the Office itself.
Maybe it refers to his obstinacy, which we have yet to see...
He could still be Peter the Roman.
I was struck by the number of times those who know him, refered to him as solid.