Jesus Christ gave the keys to the kingdom to Peter. The Rock on which that kingdom church would be built was Peter’s confession: ‘Thou art the Christ, the Son of the living God.’
This is the same foundation that Paul establishes the body of Christ upon.
Although the foundation is the same, the churches had to be different. If they were the same church, Paul would have been a liar.
Seeing the differences in these churches is vital to understanding scripture.