The office of Pope was created by the same people who wrote the Alexandrian texts of the Bible, more or less. It was an earthly office designed to establish a command structure to a church. However, Peter was never named Pope, nor was anyone else until several centuries after Christ.
The belief that Peter was Pope springs from several misconceptions. First, Jesus calls Peter a stone (Dcephas), then indicates himself (Jesus) when He says that “upon this rock I will build my church, and the gates of hell shall not prevail against it.” People mistakenly think that Jesus is calling Peter a “rock,” but He isn’t. He deliberately calls Peter a “stone” to show the difference. Each of us can be used of Christ, but only Jesus is big enough and solid enough to be the main support.
Also, Peter is “given the keys to Heaven,” in one verse. But, this seems to be a generic thing for all followers – Peter is just one of any of us who has the power to pray and bind or loose things.
Not only that, but the very fact that there is an office of “pope” goes against teachings in Scripture. When James and John were debating about which would be greater, Jesus scolded them, saying that the greatest on earth will be least in Heaven. That means that aspiring to earthly greatness (like attaining the position of pope) is a sign of wanting personal glory, and not the glory of Christ, to go forth. Peter would never have accepted such an office.