As explained, x=x is a stipulation. As such it is not the sort of thing that can be false. To suppose it false or to ask it to be justified is to misunderstand the use of “=” or of “x”.
If an axiom is understood as a substantive presumption that might have turned out to be false, then x = x is not an axiom in that sense. Nor is it properly described as an assumption. Rather, it is constitutive of the symbolism itself: once “=” is introduced as the identity relation, and “x” as a variable, x = x is fixed by those conventions. It is not a claim that stands in need of evidence or justification, but part of what makes the symbolism function as it does.