About the harmonics being in tune...
I'm not sure that's always exactly the case, even though it seems obvious from basic theory of vibrating strings.
Pianos are tuned using 'stretched octaves', meaning that bottom C fundamental is a little less than half the frequency of the C above it. I think this is because of the effect of harmonics (which may be more dominant in the perception of pitch) being slightly sharper than the a simple fraction multiple of the fundamental frequency. I guess this is to do with the string (particularly very thick piano strings) being constrained at each each, so that they are not free to pivot about the nodes freely.
I don't know how significant this effect is on a bass guitar - not very, probably!
It might well be the equal temperament error that you can hear. This is 13 cents for a major 3rd, i.e. equal temperament gives 554Hz (=440 x 2^(4/12)) instead of the 'harmonically correct' 550Hz (=440 x 5/4). Does that correspond to what the tuner tells you?