Resonance and Pitch classes
I think you gave the reason in your question. A periodic function can be written as a Fourier series, i.e. as a weighted sum of phase shifted sinusoids, where the frequencies of the individual sinusoids are integer multiples of the fundamental frequency. And the odd multiples obviously give you other pitches than the pitch corresponding to the fundamental frequency. This is what causes the same note to sound different when played on different instruments. It's called timbre.

The harmonics of the note C consist of all possible notes (and also in-between notes). The strongest harmonics with pitches different from C are G (the perfect fifth at $3f$), and E (the major third at $5f$).

EDIT: Your comment clarified your question for me. The phenomenon you're addressing is called octave equivalence. The most common argument for explaining octave equivalence is that the perfect octave is the most basic interval after the perfect unison, due to the simple relation between the harmonics of two notes an octave apart. But the assumption of octave equivalence is by no means universally accepted. As an example, every musician knows from experience that certain chord voicings which sound good in some range sound different (usually not as good) in a lower range. For some interesting thoughts on octave equivalence have a look at this page.