There are several reasons why Matthew 16:18 does not say that Jesus built the Church on Peter:
1) The Bible says clearly that Christ is the head of the church (Ephesians 1:22) and that he is the cornerstone on which the church is built (Ephesians 2:20). Ephesians 2:20 also says that the foundation is the apostles and prophets (i.e. the men through whom God gave his inspired word to the church). Note that the passage does not say Peter, but that it says all the apostles and prophets.
2) Christ's church is built on the truth that Jesus is true God whom God sent to be the Savior. This is exactly what Peter confessed when he said that Jesus is "the Son of the living God" and the Christ (Matthew 16:16). Again and again the Scriptures tell us that it is by faith in this truth that one becomes a member of the church (Ephesians 2:13, 19-20, 22).
3) It is a half truth to say that in Matthew 16:18 Peter's name (Greek = petros, masculine) and the word rock (Greek = petra, feminine) are one and the same. There is a play on words, but the words do not mean exactly the same thing. The masculine means "a stone"; the feminine means "a large rock formation such as a cliff or bedrock." Considering #1 and #2 above along with what was just said in this paragraph, the meaning of Matthew 16:18 is this: Jesus uses a play on words with Peter's name to say that the confession he had just made is the rock on which the church is built.
4) Some argue on the basis of Matthew 16:19 that Jesus gave the keys to Peter and that for this reason he is the one on whom the church was built. But just two chapters later (Matthew 18:18), Jesus says that he gives the keys to every Christian who deals with a fellow Christian the way Jesus instructed him to do it in Matthew 18:15-17. And the keys are given to any two or three Christians who are gathered in his name (Matthew 18:20). So the keys were not given just to Peter but to every Christian.