The axiom of choice is debated as a matter of if its inclusion into our mathematics produces useful math.
I don't think it's debated on the ground of if it's true or not.
And I was imprecise with language, but by saying "math is math" I meant that there are things that logically follow from the ZFC axioms. That is hard to debate or be skeptical of. The point I was driving was that it's strange to be skeptical of an axiom. You either accept it or not. Same as the parallel postulate in geometry, where you get flat geometry if you take it, and you get other geometries if you don't, like spherical or hyperbolic ones...
To give what I would consider to be a good counterargument, if one could produce an actual inconsistency with ZFC set theory that would be strong evidence that it is "wrong" to accept it.
Skepticism of a ZFC axiom in particular could just be in terms of its standard status. I don't think anyone debates that ZFC in a particular logic doesn't imply this or that, but people can get into philosophical questions about whether it is the right foundation. There are also purely mathematical reasons to care - an extra axiom may allow you to produce more useful math, but it also potentially blocks you from other interesting math by keeping you out of models where, e.g., Choice is false.