The only thing that matters is what you choose to take as granted for a particular question.
It's like how you can draw a map of a place that doesn't exist.
Or like coming up with rules for a game and then trying it to see how it plays.
Or it's like a material used for construction. You can build a house out of it, but there's no inherent "houseness" to it, despite how common such a use is.
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.
Are there essentially two flavours:
- the maths based on axioms that are fundamentally, cosmically true such as x = x. And in doing so we’re formalizing on paper the universal truths and all subsequent rules we know to be true given these
- the maths that incorporate those plus additional axioms that aren’t necessarily fundamentally true (or maybe just not provable), but work very well at laying a foundation for further rules that build a practically useful, but not necessarily “correct” toolbox
With the latter, is it kind of a “they won’t hold up under normal conditions but if you accept these axioms, there’s interesting things you can explore and achieve?”