I know you are very bright. That is evidence of a quality British education (what is that? many here are Americans, stop using big words).

None of this is relevant.

If you get deeper into higher level logic, into modal logic, into paraconsistent logic ... then no, the Law of Identity doesn't hold.

The opposite is the case. I mean, for starters, modal logic can't even make sense without the law of identity. Something is what it is. A=A. Modal logic talks of possible worlds and also of how some worlds are logically impossible... but logical possibility and logical impossibility can only hold as concepts if they can hold as concepts. if A=A.

Literally all of logic falls apart without the law of identity.

The question about axioms isn't absolute but relative. Under which conditions does the axiom hold, and under what conditions does it not hold.

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For any axiom to exist at all an axiom has to be an axiom. No other laws can make any sense without the law of identity. Without the law of identity all there is is illogical equivocation and confusion. And even that would prove the law of identity because if it's true to say that there's illogicality then it's true to say that there's illogicality... literally even arguing against the law of identity confirms the law of identity. The law of identity is the starting axiom for the whole of logic.

Cavebear is right in this way ... in Principia Mathematica, Russell and Whitehead took 350 pages of proof to show that 1 + 1 = 2. I am not impressed.

Whether you are impressed or not is irrelevant. And whether or not 1+1=2 can be proven or not is irrelevant to the fact that the law of identity is the foundation of logic. You can't deny it without denying it and if denying it is denying it then it is true and so your very own denying it is undeniably illogical. Any argument against the law of identity is self-defeating because to argue against it is to presuppose it.

It would seem they were pushing the rope, not yet aware (how could they) the relationship between logic an algebra. Mathematics is more dialectical than deductive ... unless you are David Hilbert.

Algebra is like symbolic logic with numbers. The fact that you are acknowledging the relationship between mathematics and logic, works in my favor, not yours. Truths don't help you to argue against my point when those truths are either irrelevant to my point or support my point rather than yours.