Relationship of lambda calculus to the rest of math

The question you are trying to ask is "What is a denotational semantics for the untyped lambda calculus?"

This is a difficult problem because, as Bjorn Kjos-Hanssen said in his answer, if you try and make variables range over elements of some set $D$ you find that you must have $D \times D \cong D$ and $D^D \cong D$. Unfortunately this implies that $D$ is the singleton set and all lambda terms must represent the same function.

Dana Scott solved the problem of giving a denotational semantics to the untyped calculus with the invention of domain theory.

In the standard set theoretical setup, a function cannot have itself as an input. This is because the rank of the function is strictly larger than that of its inputs and outputs.

https://en.m.wikipedia.org/wiki/Von_Neumann_universe

So when they say id(id)=id, it is meant in a more algebraic sense where composition is really just a kind of multiplication or binary operation.