How can the axiom of choice be called "axiom" if it is false in Cohen's model? From what I know, Cohen constructed a model that satisfies $ ZF\neg C $.
But if such a model exists, how can AC be an axiom? Wouldn't it be a contradiction to the existence of the model?
Only explanation I can think of, is that this model requires other axioms in addition to ZF, and these axioms contradict AC. Is it true?

The same idea as other answers, but a different explanation (which makes more sense to me, at least): there is no such thing as true, period. A statement can only be true or false in some model, which consists of a certain set of "original" statements whose truth is assumed, and everything provable from those statements. The "original" statements are the axioms.
So any given statement might be an axiom, or not an axiom, depending on the model. The axiom of choice, for example, is an axiom in ZFC, but you could make another model in which it is not an axiom.
