I heard an interesting argument from a colleague recently that went something like this. Whenever we are using an axiom scheme, we are essentially choosing one of the instances of this scheme, and hence, whether or not we include the axiom of choice in our axioms, we are implicitly using some kind of choice principle to choose that instance. My gut feeling is that this argument seems fishy, but also interesting, and I lack the expertise to give a good answer.
My question is whether this argument holds or not, and whether it makes a difference if the axiom scheme is uncountable. I realise that the question is somewhat vague, but I hope there can be some interesting answers anyway.