> why would the overtones go in integer multiples (I.e. be harmonic) for a fan?

The fan noise is from its own vibrations -- presumably driven by the motor. These vibrations will correspond to natural vibrating modes on the body of the vibrating object -- which could be the motor, or the chassis, or even possibly the fan blades. Whatever the shape, the natural modes will be naturally quantized into "harmonics". Those vibrating modes could have more nuanced spatial forms (eg. Bessel functions) but their temporal pattern would likely be sinusoid.