The piano analogy is incomplete. First, of all, a piano constructs sounds by combining multiple string sounds in a unique manner. But the idea behind transforms (Fourier being a particular case) is that you can take a function (“sound”) that isn’t necessarily produced by combining components and you can still decompose it into a sum of components. This decomposition is not unique in the general case as there are many different transforms yielding different results. However, from the mathematical (and i believe, quantum mechanical) standpoint, there is full equivalence between the original function and its transforms.

The other important point is that Fourier doesn’t really give you frequency and loudness. It gives you complex numbers that can be used to estimate the loudness of different frequencies. But the complex nature of the transform is somewhat more complex than that (accidental pun).

A fun fact. The Heisenberg uncertainty principle can be viewed as the direct consequence of the nature of the Fourier transform. In other words, it is not an unexplained natural wonder but rather a mathematical inevitability. I only wish we could say the same about the rest of quantum theory!