There are projective sets in descriptive set theory. For them, the axiom of determinacy is not contradicting the axiom of choice.
Given the axiom of choice, every set is a projective object. But in a world without AC, is any projective set a projective object in the sense of category theory? Or did those terms happen to use the same word and there is any counterexample?
It's just a coincidence. Being projective in the descriptive set theory sense is a property of a subset of a Polish space (i.e., how it sits inside the ambient space matters). But projectivity in the category theory sense is preserved by isomorphisms. So if $X$ and $Y$ have the same cardinality, $X$ is a projective object iff $Y$ is, regardless of any ambient space.
Moreover, in the category of sets, every set is projective. (Note that the axiom of choice is required to prove this.)