Is there a missing "deterministically" in the axiom of choice statement?

I can imagine the common-sense "Of course you can choose one element" assumption breaks down if we add the constraint that you need to in some way be able to have another actor choose the same element. Because for an uncountably-infinite-sized set of uncountably-infinite-sized sets, perhaps there is no way to label individual elements such that if I choose A, someone else can also choose A (how will we know they're the same A?).