The words Jesus uses when he renames Simon as Peter shows who Christ considers the OT shadow of the Pope. In Matthew 16, Jesus says:
15 He said to them, “But who do you say that I am?” 16 Simon Peter replied, “You are the Christ, the Son of the living God.” 17 And Jesus answered him, “Blessed are you, Simon Bar-Jona! For flesh and blood has not revealed this to you, but my Father who is in heaven. 18 And I tell you, you are Peter, and on this rockI will build my church, and the gates of Hell shall not prevail against it. 19I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.”
This was a reference to Isaiah 22:22:
In that day I will call my servant Eli′akim the son of Hilki′ah, 21 and I will clothe him with your robe, and will bind your girdle on him, and will commit your authority to his hand; and he shall be a father to the inhabitants of Jerusalem and to the house of Judah. 22 **And I will place on his shoulder the key of the house of David; he shall open, and none shall shut; and he shall shut, and none shall open. ** 23 And I will fasten him like a peg in a sure place, and he will become a throne of honor to his father’s house. 24 And they will hang on him the whole weight of his father’s house
Jesus’s listeners were Jews, and they knew their Torah and Prophets like we know Hollywood movie star’s personal lives. They knew it backwards and forwards. When they heard Jesus says this, they would immediately think of Isaiah which references Eli’akim, the Prime Minister of the earthly Davidic Kingdom. The Prime Minister had the authority to manage the kingdom in the King’s name. How did people know who the Prime Minister was? He had the keys of the kingdom to open and shut gates, stores of goods, armories, etc… IOW, he had the authority of the King to make decisions that were held as binding to the citizens.
Oftentimes, in NT theology, we see how Christ took earthly ideas and cast them in a new, eternal, spiritual way. The curing of physical disease is a symbol of Christ wanting to heal us spiritually of sin. The battles in the OT against foreign nations were like spiritual battles against devils. Feeding people with fish and bread symbolized Christ giving us his grace to nourish our souls. Same thing with the Prime Minister of the earthly kingdom. Peter is not a legal, military leader. He is a spiritual Prime Minister whose authority is over the subjects of the Heavenly Kingdom on earth. The Popes throughout the ages have been the successors to St. Peter and have carried on that authority the Christ handed to Peter.