The original statement of the axiom I heard was "it's possible to make an infinite number of arbitrary choices" - I think that's a good word for what you're getting at. Of course it doesn't matter which you choose - the axiom doesn't tell you anything about which you can choose - but whether you can choose at all is important. E.g. Hilbert's basis theorem relies on the idea that given an infinite-dimensional space you can always "start somewhere" - it doesn't matter specifically where, but it matters that you can pick a specific point in the space to start from.

(The first journal he submitted it to rejected the paper with the comment "this is not mathematics, this is theology")