I like the working definition "set of identifiers" because it frames identify an externalized bunch of labels, understood as relational propositions - truth uncertain - distinct from the purely internal concept of "self".
Works in a non-digital setting too: "name? dob? employer? nationality?"
If I say: "Look over there, it's Elizabeth II, Queen of England", I am establishing the sameness of a person referenced by in three different ways:
by a geographical position, by a name, by a title (according to the arbitrary rules of British monarchy)
But in the more usual sense, if one wishes to evaluate the truth of that statement, where do you start? The "over there" maybe establishes the referent to a person. The name and title then clarify a commonly understood identity for the person. We can then decide if it is, in fact, that person.
But say we already knew that we were looking for that specific person, the Queen of England. Then, the only fact we're trying to evaluate is the current location of Elizabeth the Second. Is it over there? Why yes, yes it is. Hellooooooo!
The mathematical concept of identity is not nearly that simple. Under multiplication, the identity element of the integers is the number 1. And many mathematical expressions share equivalence relationships, which is more in line with your notion of sameness.
Unfortunately (or perhaps fortunately), we can't rigorously encode the relationships you've described with any known mathematical techniques.