Is there more than one instance of the Empty Set? It seems any additional instance would be equivalent in every respect to the first, hence indistinguishable, and arguably identical. 
I.e., there is only one Empty Set.  Correct?

In the most popular version of axiomatic set theory, ZF, there is an axiom, the Axiom of Extensionality, which says if two sets have the same elements they are equal. 
However, there are non-extensional set theories. These are much more a minority taste. You will find some information under urelemente.
