Back when I owned 18 different basses, a friend with a decent home studio and I recorded me playing the same riffs on all of them (in different ways) directly into the board. The sample included passive, active, electric upright, ABG, 4 and 5 string, most major styles (P, J, stingray, Ric, T bird, etc.) and of course, a wide variety of woods.
My main takeaway was that how/where the bass was played (fingers/pick and how close to the bridge or neck), the type of string (round or flat) and the type/location of the pickup/electronics were the main drivers by far. For example, basses with unique pickups or other construction considerations like the ABG, the electric upright, the Ric with horseshoe soloed, the stingray with its active preamp, the 12 string (octave courses), etc. had distinctive sounds. Similar basses, one with rounds and one with flats, sounded really different even when played the same way. Even a passive P bass with flats can have a pretty wide sound spectrum depending on how and where you play it and where you set the tone knob.
This was my experience - if others see big differences in tone woods, etc. then they probably have better ears than I do. I don't necessarily disagree (certain woods are definitely chosen over others for legit, non-economic reasons).
Another thought is really around how clean of a sound you run. I'm pretty clean and use a DI and minimal effects (mostly just gain and EQ). As such, I find regularly playing a passive P with flats works well for me in terms of getting the sound to fit a space, sitting in the mix, etc. On the other hand, if you run through a ton of effects and distortion, those changes to your sound will be the most influential by far, regardless of what bass you use, much less what tone wood you have. Generally, whatever touches your sound last will have the most impact on it.
Lastly, you have to think about where you sit in the mix. I loved playing a 12 string by myself or with a single guitarist who took really long solos. However, the 12 string would not work when playing in some bands with coworkers where there were 3 or more guitar players (that would be a lot of mid sound).