Everyone should ask themselves: Did Simon Peter know his Scriptures? Did he know the prophecy regarding Eliakim in Isa 22? I think he probably did.
So, having knowledge of Isa 22 and the language used there, he probably took Matt 16:18 the same way Catholics do today: That Christ was giving him a special kind of authority, authority over His Kingdom on Earth. That's what the "keys", as symbols meant then (in the prophecy in Isa 22). It's about time we start realizing that it's clearly what He intended then AND now.
" I will clothe him with your robe and fasten your sash around him and hand your authority over to him. He will be a father to those who live in Jerusalem and to the people of Judah. 22 I will place on his shoulder the key to the house of David; what he opens no one can shut, and what he shuts no one can open."
"And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it. 19 I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven"
Isa 22:21-22 cf Matt 16:18