How is that different from claiming, say, there's an integer betwen three and four, oh we can't construct it but that's not a problem, gonna use it in the proofs anyway?
You can assume there exists a number between 3 and 4 though, no problems there.
As for the integer example, the integers are all easily (finitely) constructed (successors to 0 and their additive opposites), so it isn't a consistent concept to say "This is an unconstructable integer." Sets, on the other hand, are not necessarily always constructable, unless you only take the constructivist axioms.
This is where the axiom of choice is interesting; many people feel that it seems intuitively reasonable, but the implications are counterintuitive (e.g. banach-tarski)
It's different because there's nothing that says such a choice function _cannot_ exist (short of an axiom that says explicitly so).