It’s a matter of scope. Axioms are assumptions that undergird the formal edifice (those or mathematics or arithmetic) whereas assumptions are local to the theory. ‘Assumptions’ is a synonym of ‘postulates’. Axioms are foundational, such as m×n=m×n (which is true for scalars, but not true in general for matrices).
Did you mean m×n=n×m?
Both are axioms. a = a (and by extension mxn = mxn) is another axiom.
It's a distinction without a real difference. Calling an "assumption local to a theory" an axiom is fine. We have axiomatizations of both general relativity and quantum mechanics, for instance.