The first set is the rules of logic, which applies in non math stuff too. From there you can add new axioms for boolean logic, for integers, for sets, for geometry, whatever.

Even given that separation (which itself is fuzzy), even the first group can't be construed as "universally true". For example there's a range of opinion around the law of the excluded middle (whether "P is not false" implies "P is true"). Most other propositional logic axioms like modus ponens are less controversial though.

As far as real world math, while ZF set theory axioms are generally viewed as the "foundation", that's due to convention more than any real primacy of ZF. Other set theories and types of "foundations" exist that seem to be just as suitable to be called a "foundation", and most math is "foundation" agnostic. Like, if all you're doing is something with prime numbers, then it doesn't matter what "foundation" you use; so long as it lets you define prime numbers, that's all that matters.

"Foundations" only come into play when you're doing really subtle things with different orders of infinities. And then, yes, the answer can be different depending on what foundation you choose. And that's fine. It doesn't mean that either foundation is wrong. They're just different. Which is why "foundation of mathematics" is a bit of a misnomer / fool's errand. Different foundations have different results on certain edge cases, and those are interesting to investigate, not something to be upset about. And like I said, most "ordinary" math is pretty foundation agnostic.