When you pluck a string, you send 2 little pulses up and down the string - one goes toward the bridge, the other towards the nut, or if you've fretted a note, towards that fret. Sorry if this is hard to visualize - in Physics class we used to use high speed cameras or stroboscopes to see these things, so to me they're relatively easily to visualize. Anyway, whenever one of those pulses crosses the pickup on the way down the string (towards the bridge), it also (a specific time later) comes back from the bridge (in inverted polarity) and crosses the pickup again. No matter what fret you're using, given that the string is at a fixed tension, the time between those pulses is fixed. The two pulses form an impulse response, which is... a filter. A comb filter to be precise, because its a filter who's dips (and peaks) repeat every so many Hz. That filter depends on the propagation velocity of the string (which depends on its tension and linear mass density) and the position of the pickup(s). That comb filter does not depend on what fret is chosen, so it stays "pretty well fixed". Choosing a different fret changes the overall round trip time of the pulses, and therefore the pitch you hear, but the comb filter that determines the content of the harmonics that you hear does not change when you change to a different fret. It changes when you change strings, as the propagation velocity on different strings is different (It has to be if you want them to give you different notes).
Does this matter in anything more than a theoretical sense? Well, if I put a second pickup on the bass,and combine the output of both, I add impulses to the impulse response, which changes the comb filter. Does changing the resultant comb filter make a difference? People discuss this all the time here - a P has one pickup, and therefore a specific impulse response/comb filter. A J has, 2, so the impulse response/comb filter is different. If you play a P and J (both pickups on) and don't hear a difference, then.... it doesn't matter. If you do hear a difference, then yes, it matters. People her get excited about the change of position of the bridge pickup on a J - what that does is change the impulse response a bit (the impulses from the bridge pickup are closer together in time), and the comb filter is different as a result.