Category theory can be viewed as a framework to make math more modular and compatible. For example topos theory can be used to transport definitions to other "mathematical universes" of sorts

Another point of view is that is working with synthetic mathematics which is a bit like "programming" mathematics with an abstract API. The opposite, analytic mathematics, leak implementation details (for example, the von Neumann construction of natural numbers, which is analytic, leak the implementation detail that 0 is defined as the empty set; but the peano axioms are synthetic and doesn't leak any such details)

There is also the principle that two mathematical objects that are isomorphic (have the same structure) are actually the same thing. For example if you construct the natural numbers two different ways, they are "the same thing" in a sense. This usually is stated informally, but a rigorous formulation of that is the univalent principle