Is this even a meaningful distinction?


Is there a more appropriate term than "isomorphism class"?


Is there something different about a theory of sets with a nontrivial set-isomorphism class of empty sets?



Are there any consequences of the initial object in Set being unique, while the isomorphism class of terminal objects is nontrivial?
The distinction is meaningful in many senses. Certainly it is on ZF foundations, and contrary to what Dylan Wilson might seem to suggest, the distinction can be stated formally in the Elementary Theory of the Category of Sets as well as in the standard axioms for the Category of Categories as Foundation. 


I do not see any reason to use any term other than "isomorphism class." That is a vague question.


As to the third question, there is no important difference. You know you could require just one terminal set. In ETCS or CCAF you could add a skeletal axiom saying there is just one set in each isomorphism class. And on ZF foundations even without changing the definition of function you could pick one singleton and insist it is the only singleton allowed in your category. It would not achieve anything important -- as you can sort of see from the triviality yet artificiality of the ZF way of doing it.


