In the usual development of axiomatic set theory we use sets to represent all of the mathematical objects we want to reason about, including numbers. This is a choice, not something that inherently has to be so.
An alternative would be to develop the axioms of set theory in a way that allows urelements -- things that can be elements of sets, but are not themselves sets -- and we could then extend the theory with axioms that assert that some of these urelements happen to behave like numbers; that there is a set of all numbers and sets that represent the addition and multiplication functions and so forth.
It is not usually done that way, because the main interest in writing down completely formal sets of axioms is to investigate the limits of what can be proved at all rather than conduct particular proofs within the system. For this purpose it is technically convenient to have as few axioms as we can get away with -- so given that the purely set-theoretic axioms turn out to be enough to build everything else on top of when we need it, it would just be dead weight to carry around additional axioms for numbers in particular.