Craig Burton has been generously heaping kudos my way for aruging that a discussion of the philosophy of identity is orthogonal to the discussion we should have about the laws of identity. And I'm going to hold firmly to that direction, even though I received this titillating comment – probably more relevant to Scott Lemon's Axioms – from David Rollow of CSG Systems.
Take a look at W. V. O. Quine's essay “Two Dogmas of Empiricism” for relevant philosophical discussion about the most basic identity issues. If you understand it you will realize that the very idea of “axioms of identity” makes no sense. The “personal” identity discussion in philosophy is about things like persistence, sameness of tokens observed at different times, self-knowledge and self-identity, problems of no interest in the age of identity theft. The Quine essay is at a more fundamental level of discussion but it applies if you apply it.
Yes, the “it applies if you apply it” view of philosophy is hard to argue with, especially if you argue with it.