To be maximally pedantic, sine waves (or complex exponentials through Euler's formula), ARE special because they're the eigenfunctions of linear time-invariant systems. For anybody reading this without a linear algebra background, this just means using sine waves often makes your math a lot less disgusting when representing a broad class of useful mathematical models.

Which to your point: You're absolutely correct that you can use a bunch of different sets of functions for your decomposition. Linear algebra just says that you might as well use the most convenient one!