The axiom of choice is not needed to choose from one pair of socks. You do that every morning when you put on your socks. In fact, even if you have infinitely many socks, choosing one doesn't require the axiom of choice. Neither does choosing five, or ten socks. That's just existential instantiation and induction.
The axiom of choice is needed in order to choose from infinitely many pairs at once. The reason is that between two socks, there is no "left" or "right", and there is no distinguishing property that we can always say that "given a pair of socks, one of them is such and such". And so the axiom of choice is really needed for that.
More formally, Fraenkel, and later Cohen, showed that this argument is mathematically solid. It is consistent that the axiom of choice fails, and there is a set which is the countable union of pairs, but there is no function which chooses a single element from each pair.