Reading through the tag wikis for set-theory and elementary-set-theory suggests that the axiom of choice is improperly included in the former when it should be in the latter. Note that elementary-set-theory already includes cardinal arithmetic.
I am not suggesting that applications of the axiom of choice are necessarily, or even usually, easy or suitable for high school students or beginning undergrads. But I think we can and should base these tags more on the difficulty of the concepts involved than on the difficulty of the proofs involved. An undergrad can read a difficult proof using the axiom of choice and understand each step, but may not grasp the structure of the whole. She will not, however, be likely to make head or tail of a forcing argument.
The tag excerpt for set theory:
This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.
The tag wiki for elementary-set-theory:
This tag is for elementary questions on set theory—the sort of material covered in undergraduate set theory texts. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, countability and uncountability, power sets, transfinite induction, well-orders, ordinal and cardinal arithmetic, etc. More advanced topics should use the "set-theory" tag instead.