This is just math, not physics. Suppose you have a thing vibrating in a periodic manner. You might imagine that it will vibrate the air so that the sound pressure is some periodic function of time. Fourier transform that function to get a spectrum, and it will have discrete peaks at the fundamental frequency (1 / period) and at integer multiples of that frequency. You don’t even need to decompose into sine and cosine functions for this to work — all that’s really going on is that you have f(t) = f(t + period), and you’re turning f into the sum of a bunch of other functions g_1, g_2, etc, all of which have the same property that g_i(t) = g_i(t + period). Of course, if a function g has the property that, for all t, g(t) = g(t + period/n) for any integer n, then you can iterate that property n times and you’ll also have g(t) = g(t + period). And these functions with the fundamental period, half the fundamental period, one third the fundamental period, etc, are the fundamental tone and its overtones. You could decompose into square waves or just about anything else and you would get the same result.

(In any discussion of Fourier transforms complete with equations, you’ll usually see a bunch of factors of 2π because the frequencies are angular frequencies. This is done for mathematical convenience and has no effect on any of this.)