If it was an acoustic effect, I would expect a difference in sustain, also.
If not, i would say it's all in the pickup.
A heavier string (the E) will normally produce higher output for the same
distance from the pickup. So that one bass sounds normal. Typically, you
lower the pickup on the low string side. E and G pickup gaps are usually not
equal. You do need to set the pickup heights on each and every bass.
Not just for balance, but also for good tone.
The jazz bass sounds unusual, but I can check mine as soon as I get a chance
and see how the pickup gaps are set.
This is a bit of a crude test, but you can touch the tip of a screwdriver or small
steel rod to each of the pickup pole pieces, and estimate their strength by how
hard it is to pull the steel off of the pole piece. They should all feel the same.