Your question displays sufficient knowledgeability of the phenomenon of strictly integer harmonics, that you can basically disregard the replies explaining how they could arise, whereas your question is more about given that it shouldn't arise inevitably, how do we know they would be harmonic given the case of a fan.

Some of the replies pontificate and assume sounds are periodic, and hence their harmonics must have been perfectly integral, which is of course totally bonkers.

Yes, some instruments are harmonic (i.e. integral harmonics down to ~ ppm frequency ratio errors) like violins, but only because those are bowed strings, resulting in phase locking.

Plucked strings are much further from integral harmonics, due to dispersion: yes standing waves for a frequency-independent wavespeed c on the string would give perfectly harmonic partials. Real strings show dispersion (a frequency dependent wavespeed) resulting in inharmonic partials.

Nothing indicates fan noise to be strongly harmonic. Their composite sound may have structured and repeatable (in)harmonic components in many ways, harmonics would be easiest to explain. The part that sounds "white" would presumably be hard to characterize and cancel.