The axiom of choice is about making an infinite number of arbitrary choices simultaneously. If you can specify some rule, this rule is just one choice. Finitely many choices are always fine and don't need the axiom of choice.
If you have a finite set, you can number its elements and make rules by saying "let's take the element with the smallest number having this or that property", so you don't need the axiom of choice when dealing with finite sets.
The point of Russell's shoes versus socks analogy is that shoes are distinguishable while socks aren't:
To choose one shoe from each of an infinite set of pairs of shoes, you can always choose the left shoe, or specify some pattern (so you don't need the axiom of choice), whereas when choosing socks, you have to make an arbitrary choice to select one from each pair (so you do need the axiom of choice).
If you have a finite set, you can number its elements and make rules by saying "let's take the element with the smallest number having this or that property", so you don't need the axiom of choice when dealing with finite sets.
The point of Russell's shoes versus socks analogy is that shoes are distinguishable while socks aren't: To choose one shoe from each of an infinite set of pairs of shoes, you can always choose the left shoe, or specify some pattern (so you don't need the axiom of choice), whereas when choosing socks, you have to make an arbitrary choice to select one from each pair (so you do need the axiom of choice).