Help!
His argument basically goes that two identical spheres in an otherwise empty universe are indiscernible, but not identical. He also mentions the axiom of choice in passing. This got me thinking about the choice function for finite sets: does it even make sense to say that we can select an element x from finite S if x is indiscernible? Does x even exist as a choice if it is indiscernible from the other elements of the set (but unidentical)? It almost seems for a second that it has become necessary to invoke the axiom of choice for finite sets.
After doing more research, I think what it boils down to is: in regular ZF set theory and most set theories built on classical logic, the ability to choose from a finite set is guaranteed from the rules of logic only (not requiring the axiom of choice). So in that scenario my question is nonsense.
However, we can enter a paradigm where we reject that a choice function exists for e.g. the two spheres scenario. This is constructive mathematics, where we reject one of the rules of classical logic (the excluded middle). We can then develop a new set theory according to that new logic system: https://en.wikipedia.org/wiki/Constructive_set_theory.
Thanks @Karl for helping point me in the right direction.