This is a beginner's question about foundational construction of (alternative?) number theory. The notion of mathematical equality is closely related to logico-philosophical notion of 'Law of Identity', and Peano axioms start from defining equality (axioms 1-4 according to wiki article) with a certain logical apparatus, which presupposes law of identity, and only then proceed to "Successor function" of additive property.
The basic intuition here is that it could and imho would be more natural to derive mathematical equivalence from more basic "countable" or "ordinal" relations, formally relational operators < and >, than from formal presupposition of Law of Identity which basically gives "=" as given axiom. Deriving equality from "neither less nor more", e.g. if not not equal "<>" then equal "=" seems better connected with universals of counting and ordinality in natural languages of the world (AFAIK at least ordered finite set 'one', 'two', 'three', 'many'/'more' can be found in all languages) than the Peano approach.
Wiki article on relational operators indeed states that "relational operators can be designed to have logical equivalence, such that they can all be defined in terms of one another". If I'm not mistaken, the "cardinal" aspect of natural numbers, which arithmetic functions seemingly requires, could then be derived from relational equivalence in consistent manner. and cardinal numbers as relational identities would be subcategory of more fundamental ordinal relations < and >, and further relational operators derived from those.
I won't attempt to fully formalize this idea here, but leave the task open for any takers. I don't yet know how this idea relates e.g. to Skolem's approach, Löwenheim–Skolem theorem and Skolem's paradox, ie., how dependent those are from the logical apparatus and set theoretical approach used, and do their results extend to mathematics more generally, and welcome all contributions.