I'm beginning to think it does not. I think the individual construction and the wood itself has everything to do with it. Like the Kerry Green one: It literally had 1/8th of wood sanded from every inch of it, top, bottom, back of neck headstock front and back...But I put 17 coats of PPG lacquer (mainly because I was new and forgot about grain filler!!!) and even though this bass was thin, it played like a dream!!!!
As for the white one...I had the neck reset, painted it with old school Dupont Polaris White lacquer Lucite. boyt 6-7 coats. It sounds killer...but not like the Kerry green one.
Now, as some of you may know, I only own 2 Thunderbirds, 2 1965 Thunderbird II's. One is refin Pelham Blue and the other original.
The difference I have found in both of these is...The original sunburst one is somewhat thinner in the neck, not as beefy as th blue one. It has a Great tone...But the refin Pelhem Blue one has a slightly beefier neck and sounds like a P Bass on Steroids!!!! It makes no sense.
As for pickups differences...In both the Kerry Green Non-Reverse and the Refin Pelham Blue reverse II, I've tried several pickups.....Non really making a difference...It's the individual bass for sure...the wood...where it was grown...humidity factors....tolerances in construction...who constructed the individual bass at Gibson..etc....