IME, wood and construction do have some impact on how a bass sounds.
I have had four Precision-style basses, and they all responded in a different way, also when unplugged. One was an Am Std Fender, maple/maple over alder. Quite heavy but not too heavy, a modern quite thin neck. Another was a Yamaha BB Billy Sheehan model, maple/rosewood over alder. Substantial weight and a *thick* neck. The third one a cheap Vintage by Wilkinson, maple/"rosewood" over "alder" (note the quotation marks). Lightweight and with a medium-thickness neck. The fourth one was an Orange O-Bass, which has the pickup slightly towards the neck (8mm, not much really). Maple/ebony over okoumé (similar to mahogany but sturdier, apparently). A very wide neck. I installed the same pickups (EMG GZR) in the Fender and the Vintage. The Orange had a Seymour Duncan SPB-1.
The Yamaha was quite bright and with pronounced mids around 300-400Hz. It sounded best with a bit of a bass boost and a slight mid cut. (If that inherent mid boost had been an octave lower, I would probably have been very happy). I'm talking about the split coil pickup alone, blending in the big neck pickup made it sound a bit closer to what I liked. It had a bit of a "stiff" character (not bad, but I wanted something rounder and with more bloom, especially pickstyle).
The Fender was very bright, with very pronounced upper mids. I needed to cut the high mids on my amp with this bass because otherwise it sounded a bit too clanky. The bass vibrated very little when I played it, compared to my other two Precision-styles. It had a very "precise" character (well, it's a Precision...
). I felt the bass sounded a bit too clinical for my tastes, no matter what strings I'd use. I didn't experiment too much with it because I sold it to buy another bass.
The Vintage by Wilkinson has an uncomfortable neck profile and it has a slightly smaller body than a regular Precision (but with large '70s style tuners). It has a little bit of a neck dive. It is my favourite so far
it sounds the closest to what I want it to sound (by the way, I saw very little variation between the original Wilkinson pickup and the EMG). It is the bass that vibrates (physically) the most. It's less "precise" sounding than either of the former two, but it's the sweetest sounding nevertheless. Now with a set of DR Fat Beams.
The O-Bass... I couldn't get it to sound right with roundwounds. Like, at all. It sounded a bit hollow, thin. I first had very good impressions with Fender flats - the most "consistent" tone I've ever had out of a bass, if a bit unexciting. But it really started to shine with Pyramid flats. It was always a bit lean in the bass and it didn't really "fill" the mix, but it had this fantastic mid tone that was very nice both finger and pick style. I then sold it to a singer-songwriter I play with because I have too many basses
Now, pickup and pickup placement is key. Having a split coil (it doesn't really matter which, they all sound similar, especially if they use the same kind of magnets) exactly where it is (±8mm...) gives the bass that kind of timbre. But how that timbre actually translated into the mix was very different between the four basses. I have no idea why. Nevertheless, each bass actually sounded different unplugged (with the same strings, of course).
And these are all solid body basses. Hollow- or semi-hollowbodies are another thing altogether, even the ones with a centre block (the hollow wings do give them some je ne sais quoi).
One thing I have noticed: I don't seem to particularly care for heavy basses tone-wise. I used to have a Peavey T-40. It didn't sound particularly bad, but it had this prominent midrange honk (in any of the pickup configurations) that I had to dial out (and even then, it tended to sound a bit "cold"). I saw similar behaviour (both plugged and unplugged) between the Peavey and the American Fender, albeit more extreme in the Peavey.
My favourite bass to play? My completely hollow Chinese Höfner. It's crap and I love it.
(My experience as regards strings is that flats have much more variability than rounds. Roundwounds come in three types: nickel, steel and Rotosounds. Gauges scale similarly across brands. All the rest is made up of minor variations, string life, etc.).
