It's a mathematically correct way of looking at it as long as you're clear that that the bases are functions, and you're considering how a particular mathematical artifact (the signal function) is represented as a sum of basis functions. In the naive signal trace, the basis functions are one-hot functions (zero everywhere except at a single point, where they have value 1), in the Fourier basis, the basis functions are sine waves of amplitude 1. So the signal is the sum of the basis functions multiplied by a unique amplitude for each basis function.

Sine waves aren't the only basis with which you can make this transformation. Your basis can be an arbitrary set of periodic functions as long as it meets certain requirements. Decomposition into wavelet functions is commonly used in seismic signal analysis, for example.