People question C all the time. That might be the most prominent ideological difference in mathematical philosophy.
Does it matter? Of course not, but people question it anyway.
For example the axiom of infinity (by finitists), the power set axiom and first-order theories in general (the downward Löwenheim-Skolem theorem implies that the infinity and power set axioms can't guarantee the existence on an uncountable power set), the fact that ZF doesn't allow a set of everything, and in particular no proper set complements, the fact that the axiom of regularity seems to be useless, etc.
Of course most ordinary mathematicians don't care about all that, because they don't care about ZF(C) or set theory or the foundation of mathematics in general. They rather care about problems in their specific field, like algebraic topology or whatnot.
Comparing the axioms of math to relativity in physics is just nonsensical. Math is independent of observation, if a proof is formally correct now, it will always be correct under that chosen axiomatic system. Sure, we can play with different axioms (as others commented, it's common to drop the axiom of choice), but that doesn't invalidate the previous work at all.
The concepts such as true, false, equal, greater than - all refer to human experience with counting things or perception of existence etc.