According to the time-binding record, "A=A" is the Law of Identity, just one of the Laws of Thought. Although it is attributed to Aristotle, it is not what Aristotle ever said.

The "law of identity" as it has evolved and been passed down to us concerns logic and mathematics. It is really not about what "is". The closest expression of that that "is" about what is is "whatever is, is", and is about being qua being.

The formulationa "A=A" and "S(A)=S(A)" have two very different meanings. In "A=A" the traditional "universe of discourse" is logic and syntax; it is not about semantic relations between symbols and referents. The formulation "S(A)=S(A)" can be interpreted to mean that the "structure" of "A" is the same as the "structure" of "A". But this is does not readily represent two different levels of abstraction as when we speak of the water on the one hand and the river on the other hand. Aristotle's On Generation and Corruption discusses the nature of change in terms of two distinct levels of abstraction.

In traditional philosophy "identity" speaks to what makes a thing such a thing, and this goes back to Platonic ideals and Aristotelian essences. If we are to try to express something like this in a language remotely consistent with general semantics, we might suggest that such an "essence" is a "highest" level abstraction such that nothing that is not always attributed to such a thing is included, and everything that is always attributed to such a thing is included. It has been argued that there is one and only one characteristic that fits this model, and that is, in clasical philosophical terms, its "essence". Examples of such abstractions include "chairness", "dogness", etc. With more than twenty-five centuries of common useage of this paradigm, that usage is still ubiquitous on the face of this planet, and it permeates our language. The law of identity then holds that everything has such a unique single characteristic, and that it is unchanging. It is pure, and no matter how abstracted, by whom it is arrived at, it will be the same. That which is is what it is. Whatever is, is. It also seems to me that a great many people confuse the "identity" relation, which is an equivalence class with one member for each member in a set, expressed by A=A, is quite different from saying that A is A. When we say A is A, we are going back to ancient forms which predicates a property on a subject. Subject A has property A. In this case the subect A names what is to be talked about, and the predicate A identifies the essential property that is unique to the named suject and only the named subject. But because we are talking about the "essence" or "essential" quality, we cannot use any other formulation, and there's the rub. We see "the same" symbol in two diferent places, but tend to forget that they indicate different meanings.

I submit that these multiple meanings are thouroughly confused among the lay population and many others.

With general semantics we are not likely to "believe in" "essences" as being anything other than a fuzzy and ill defined hypothesized high level abstraction. We are also not likely to believe that such high level abstraction can be "unique"; that is to say, we might think that even the "highest" level abstractions have multiple possible characteristics. (This could be an arguable 'nother topic.)

If something has a "degree of differentiation", it has a number of component parts.

Tisk, Tisk, Tisk. Does every part have parts? (All the way down?) Do we have "infinite regress"? Not according to our current model of physics. At the bottom are "quarks".

If the degree of differentiation and the organization are the same at all levels, then, at that instant, the structures are "identical" except for their time-space coordinates (location in space-time). Measuring this, however, is subject to the Heisenberg uncertainty principle, so we cannot, in principle, determine identity by measurment.

We must not confuse "existence" with "knowledge".
We cannot "know" if our criteria for identity can ever in principle be met because of uncertainty in measurement. Therefore we cannot "know" if identity can exist. It is confusing metaphysics and epistemology, existence and knowledge, to assert that "existence" of identity is impossible. Knowledge of existence of identity is what is impossible, according to our best current model of physics.

In our symbolic environements, however, "identity" "exists" by stipulation.