I believe you missed my point entirely.

"Why did your ear, the outer part and the inner bones and the canal and the fluids and the brain, all evolve together in a way that grants Fourier analysis for free? Coincidence? "

No, we model physical systems using math, so you should expect that a physical system involving dynamic systems to use Fourier transforms to effectively model that system. But the efficacy of math does not imply that mathematical objects exist.

"Maybe maths could help us understand biology or other non-maths fields better!"

Um, clearly. Math is extremely successful at describing many aspects of many sciences. But this does not imply mathematical objects exist and are describing the reality of what is occurring, despite providing accurate outputs.

Phase invariance is a side-effect of doing the transform to the frequency domain, isn't it?

It’s not a coincidence, but it is only an approximation. Fourier analysis of sound assumes a continuous field of air pressure at each point in space, which evolves according to a continuous function; in reality, air consists of a large number of discrete particles, and “pressure” approximates the result of a large number of discrete, random collisions. We can prove mathematically that the more particles are in the system, the closer its behavior gets (with overwhelming probability) to the continuous ideal; and sound happens to operate at a sufficiently large scale that the discrepancy is far too small to make a difference, so we - and evolution - can just use the continuous ideal for our calculations. But that doesn’t require it to have any physical meaning.

On the other hand, at a lower level of abstraction, the most fundamental theories of physics known do tend to involve real numbers and continuous functions – from my limited understanding, that applies even to quantum mechanics in many cases, even though it’s known for discretizing quantities that were continuous in classical mechanics. However, it’s unknown – unknowable, even, at some point – whether these infinities are “real”, or themselves approximations of even more fundamental laws.

Sure there is. I can use Fourier transforms because they are efficacious but this does not force ZFC upon my reality. Could they be apart of reality? Sure, and, if they are they produce efficacious tools. Could they not be apart of my reality and efficacious tools still be available? I don't see why not.