> I am bothered by the fact that most mathematicians aren’t bothered by that! Am I missing something?
Outside of logic and set theory, most mathematicians aren't really working in any particular foundation: they're working directly with the relevant structures. An ordered pair `(x,y)` can be modeled as the set `{{x},{x,y}}`, but that doesn't mean it is that set. What it is, to the extent the question is even coherent, is a thing that has the data ordered pairs are supposed to have: a first element, and a second element.
Outside of logic and set theory, most mathematicians aren't really working in any particular foundation: they're working directly with the relevant structures. An ordered pair `(x,y)` can be modeled as the set `{{x},{x,y}}`, but that doesn't mean it is that set. What it is, to the extent the question is even coherent, is a thing that has the data ordered pairs are supposed to have: a first element, and a second element.