Not only identity but all knowledge including physics and mathematics are simply just Tools.
A is A because it's named that way and 1+1=2 because it's designed that way. In binary systems 1+1=10
I found out that my thinking aligns with http://www.newworldencyclopedia.org/entry/Instrumentalism#Early_Instrumentalism
A physical theory is not an explanation; it is a system of mathematical propositions whose aim is to represent as simply, as completely, and as exactly as possible a whole group of experimental laws (Duhem 1954, 19).
Thus physicists need not believe that the fundamental laws of physics (e.g. Einstein’s light principle) are true or that the fundamental physical entities (e.g. electrons) actually exist. Rather, these are just useful fictions that help physicists predict phenomena.