I might be reading something into your thoughts that you're not intending, so feel free to take this with as much or as little relevance to your enquiries as you wish.
It seems like something that might be a useful piece of conceptual technology for you would be the idea of Logical Intension. In analytical philosophy of language, we often draw a distinction between two senses in which a word, a naming or predicative phrase, or a sentence means something.
On one hand, we might have a direct notion of reference - that is, the thing to which a name refers. So if I mention "the tree in the courtyard", to say of this phrase that it has a reference is to say that there is something in the world (or to avoid metaphysical readings, we might say in my frame of reference) such that it is a tree, and that in using that naming phrase, I might be intending to specifically ascribe properties or descriptions to that thing.
Of course, not every sensible piece of language I might want to use has an obvious reference. Let's say for example that I want to talk about unicorns. Now unfortunately, there are no unicorns (as far as I'm aware; at least, there certainly aren't any unicorns in my frame of reference). But on the other hand, that doesn't mean that any instance of the phrase "unicorns are..." is without meaning. I can mention unicorns, confident that other people will understand that I mean to talk about creatures like spiral-horned horses, even in the absence of anything to function as a clear referential object of the term "this or that unicorn".
This other kind of meaning of a word, a kind of indirect, descriptional notion of meaning where we associate words or phrases with some kind of psychological event, construction procedure, experimental verification or falsification scheme etc. is called intensional meaning or just the intension (or sometimes sense, in the work of Gottlob Frege who is often credited with drawing the distinction in formal analytical theory), in contrast with the "extensional meaning" that we more commonly think of as just "the stuff that we're talking about". The idea is that we draw a distinction between the domain of things that the language is used to denote or talk about and the means of presentation of the things we want to talk about that language use signifies.
So to your particular problem then. Abstractly, we might reflect on what it means for something to be a proposition or an object of identity, and thus give it a generic name. Let's say A. A names something in the world, but we might say that there could be some other description of the same object that A refers to, but that has a distinct intension from A - let's call this B. So A and B are stipulated to refer to the same thing, but that have different intensions, as noted in one of them being A and the other being B.
Now! The statements of identity that A=A and A=B look to be very different statements. To say that A has the same reference as itself is in some sense uninteresting. It seems to be little more than an intuitive sense of what it means for a relation to be an identity relation - namely, that it matches objects on either side of the = symbol (this is a sense in which the axiom of identity can be read - as specifying that A=A is something of a prerequisite for some relation to be properly considered as a candidate "identity" relation. in different contexts, 'Identity' might have different additional connotations, though I'm staying clear of that for now). Yet to say that A and B, intensionally distinct descriptions, have the same reference as each other, might be quite interesting. That is, two different ways of presenting an object in some sense converge towards one another at the level of referential identity.
One popular example might be "Clark Kent" and "Superman". It's part of the mythos of the story of Superman that he takes on the persona of Clark Kent to disguise himself on Earth, so the two are the same on a direct referential level. But actually, finding out that Clark Kent is Superman is, to the people that know either Clark or Super independently, quite a big deal! Who knew that the mild mannered guy working in the news room actually turned out to be a superhero crime fighter, right?
Drawing a distinction between extensional and intensional meaning might help you puzzle through some of your thoughts about how perspectives and axiomatic identity relations might come into play here. How do we use our language, logic and conceptual technologies in making claims and assertions about a given frame of reference, and how do matters of good practice, convention, shared reasoning patterns or communication standards influence the ways in which we connect elements of our objective picture of the world together?
Because I think you're right in many ways - on an extensional level, axiomatic specification of what it takes for something to be an identity relation doesn't tell you everything that might be psychologically or computationally interesting about what kinds of things are identical. It might be that without enough of a background framework about what kinds of things we want to include as modelling assumptions, conceptual tools, community conventions etc. for intensional descriptions, distinctions that we think should be obvious and rational just can't be drawn.
And if we think that's okay, then what's the specific motivation of excluded middle or non-contradiction? If it's not prima-facie simply logical (in the classical Aristotlean sense) that there are referentially distinct things, or that all distinctions that we take to be logical occur strictly at the level of reference, then what does the idea of Negation amount to? Why should there be neat and clear distinctions between any given property and its lack, absence or exclusion?
I think there are interesting proposed answers to those questions at least in as much as mathematics, culture, psychology or social structures are the domain of logical enquiry, though their methodologies are very much distinct, and in many ways significant (and potentially dangerous) issues emerge when we try to think of any framework as a dominating philosophical paradigm. The question that might be worth thinking about as a prerequisite for making progress here is what, or perhaps even who, your philosophy is ultimately for, and then to have a look at how they progress in their investigations and model building.
