Quarks are an example. Quarks were conceptualized as a mathematical simplification, and only later were recognized as real physical things - or at least as real as protons.
In QM we have physical quantities (an electron's mass) and unphysical quantities (an electron's phase). But there's open questions, like the wavefunction - can that be placed in correspondence with reality? We don't really know. This is the "ontic" vs "epistemic" debate.
Reminds me of tunnelling. That's one of those "it's weird but the equation should work on the other side of the barrier" situations, lo and behold stuff goes through the barrier.
For example in mechanics the coordinate systems being used aren't "real" in a physical sense because you are free to choose things as you wish. But the force you calculate is very much a physical fact, a property of the real world.
This doesn't preclude there being a better model of course. Observations can get more sophisticated, revealing that your model needed adjustment. Or parsimony can drive and someone finds a way to describe multiple phenomena at once, eg electromagnetism.
As for why mathematics is great at describing nature, it's because there's loads of it, since it's an extensible system of logic. Someone somewhere is going to find a way of describing how asteroids go round, and being a kind of logic there's no way you would say it wasn't math, regardless of what exactly it was. Epicycles are math too, despite not being the exact right way to think about it.