It's the nature of resonance and vibration.

If the fan has any recognizable pitch at all, it's because something periodic is happening. If it's loud enough to be annoying, there's probably some resonance going on to amplify it.

For example, maybe the motor spins at 120 Hz, and it's slight asymmetry shakes the chassis of the fan. That shaking will send waves across the body of the fan. Any of those waves whose wavelength is not an integer multiple of the size of the body will bounce around and end up destructively cancelling out. But the wavelengths that are at are close to integer multiples of the resonating frequency of the body will reinforce themselves as the bounce back and forth across the chassis and get amplified.

If you do an image search for "string overtones", you can get a picture of what I mean. Random physical objects aren't all strings, but many of them have at least a little plasticity and rigidity such that they can vibrate and resonate. When they do, the result will be harmonics at the object's fundamental frequency and integer multiples.

Other frequencies occur too. If you strike a bell, for example, that impulse will produce waves at basically all frequencies. It's just that the ones that don't resonate with the bell's fundamental will cancel themselves out and fade out nearly instantly (that's the clanky part of the very beginning of a bell sound). The multiples of the resonance frequency will ring out (the bell-like peal that decays slowly).