The following is my OPINION, and it is my experience, yours may vary. I know this will create some turbulence for some here.. but in my long time repairing guitars and basses, I have had the occasion to experience the same pickup in different basses. I was building a five string a long time ago now, and wanted to experiment with different pickups, placement, etc. I made a surrogate 'body' with a few pieces of maple. There was a large open area from the end of the fingerboard, to the edge of the bridge. The very first pickup I tried, was a good old P-bass pickup. Probably a Fender pickup, as this was before I had a stockpile of aftermarket ones. This was so long ago, that I did not have a low 'B' string in stock- so it was simply tested for overall tone with only four strings. I centered the halves at 11-5/8" from the center of the 12th. fret, where P-bass pickups are almost always located on fender basses and their clones. The scale was 34." That neck- is on the bass in my Avatar. It was completed in mid 1994. It has a very expensive Alembic circuit. Does it sound good? Oh, yeah.
So what did it sound like as just a neck on a block with a p-bass pickup? Exactly like P-bass. A pretty good one, too. Subsequent experiments yielded similar results using my notebook made over decades of measuring just about every model of bass that came through my shop. In my opinion, body wood, neck and fingerboard wood, are smaller parts in the tone pie chart, and I know that from my own builds.
What REALLY makes a bass sound a certain way- is the pickups, type, shape, configuration, and placement in the scale length. On that tone pie chart, I'd say that makes up about 70-80 percent of the overall tone of a bass. My other builds that use the same layout as my yellow bass, and sound remarkably similar. EMG's in one, Bartolini's in another. The placement of the pickups creates the similar tone.
P.S., I have a Warmoth build, Swamp Ash, Maple neck, P-J configuration, The P-bass pickup is from an early 70's Fender bass, and the bridge pickup, is a Schaller- that is exactly like a 70's DiMarzio hum cancelling. What does it sound like? -A Fender P-bass. A good sounding one. It sounds just about the same as my other P-J with a Maple body and a Maple neck with a rosewood board, and EMG's.