A=A can't be used as an axiom though, as it doesn't add anything of value to any logical system.
Let me rephrase that.. It *could* be used as an axiom (you can use anything, if you want), but it'd be redundant, being a tautology.
I agree, I think it's use is primarily as an "excluding principle" that any arguments of the form A=B will be rejected. As to whether it can be used as an axiom, I think you're right [but I don't know enough about logic - axiom systems can get very complicated and breakdown at a high level in formal logic, humans are far from sorting logic out as a subject].
I think Rand's point was as follows, in order to prevent duplicitious arguments of these forms:
MIND
mind = mind
non-mind = non-mind
A = A
mind = non-mind
A = B [edit: more correctly, A = non-A]
The first two arguments are ok - they are of the form A = A, but Rand really hated arguments of the third form, where it switches without warning to A = B
Same thing with arguments about reality:
REALITY
reality = reality
non-reality = non-reality
A = A
reality = non-reality
A = B [edit: more correctly, A = non-A]
First two arguments are okay - it makes sense to try to argue for reality or non-reality, but as non-reality can't be argued for successfully, people switch to A = B of the third form. These type of arguments should simply be rejected.
So the function of A = A is an excluding mechanism - don't accept arguments that use your mind to tell you that you have no mind, is the basic principle.