It’s still just a renaming. In fact you might even perform the renaming and thus form the bijection without even realizing it, if you talked about “the first square number,” “the second square number,” etc.
You need neither though. For example:
y = x ^ 2 when even, (x+1) ^ 2 otherwise
It's a bijection. (And therefore obviously also a surjection.) "One-to-one correspondence" is a correct, but incomplete, description.
Unless you consider the domain of f the "renamed" range, in which case yes tautologically it is bijective.
You seem to be talking about the function f: N -> N, which isn't surjective, but I don't see the relevance. Any surjective function can be made non-surjective by considering a larger range.
It's worse than just irrelevant. The claim is that we have two sets, ℕ and S, that are the same size. The argument that they are the same size is the observation that we've demonstrated a bijection between them.
No one ever mentioned a function from ℕ to ℕ, because that would tell you nothing about S. It would be like saying "in order to show that 5 is odd, observe that 2 and 2 are both even". The idea is gibberish.