A bit more on this... Sorry for these long posts...
I'm the first to admit that my sound palette is limited. I have great respect for those guys who can make a lot of different bass sounds but that's just never been me. I'm a one-trick pony and only know how to slightly vary the tones that I make. For instance, I know nothing about pedals or even amp tones - always just been a DI guy. I never even got into slapping and popping - I left that to the specialists.
And I understand that what I do might not work for everybody in every situation. (Mostly talking about recording as that's the bulk of my experience). Thus my comment about Alembic basses not being everybody's cup of tea. The sound these instruments make is LARGE. I find it very musical, natural, and almost like an acoustic instrument. But it's more like a piano than a "standard" electric bass (to my ear). The sound a Fender bass makes is by comparison small. The entire frequency range of a P-bass might not go as low as the kick drum and might not go as high as the guitars. But some folks want the bass to only sit in that limited frequency range so they can crank it up and it doesn't interfere with any other instruments. I acknowledge this and respect that approach, but then I may not be the right guy for that situation. All good though, to each their own.
That said, since our basses have such a big sound they can be EQ'd to do literally anything. Once I've recorded a part for somebody I'm happy to turn it over to them to work their magic on it and make it fit into their track as they see fit. Sometimes the results might make me shake my head (nasty tone compressed to death... is that still me?) but generally my experience has been good and I've made friends with a lot of great recording engineers.
Now a bit of an aside. Some people are much MUCH more attuned to the sound of particular guitars or basses. I recently got to watch Jackson Browne and his great band play several shows and the guitar trunk on stage right had 13 guitars in it. Those were all for Jackson. On stage left were Greg Leisz's 10 instruments and 12 more for Val McCallum. That's 32 different guitars. Needless to say, the two guitar techs were tuning non-stop! But they wanted very specific instruments for specific songs. As a listener, I gotta say, I don't think I could hear the difference. But it was important for them so ... whatever.
Again, it depends entirely on the situation but I think it's generally not nice to tell a musician they can't play the instrument of their choice...
Michael, I kinda think Jaco, like Louis J, would sound like Jaco on any instrument he picked up. He was known to sit in wherever he showed up and play whatever he could wrench out of the bass player's hands. It really didn't matter, there was no doubt who was playing bass.
My fretless bass is the biggest variation I have to offer sound wise. But I have had bassists I know - who should know better - ask if I was playing fretless on a specific track which I was not. So I guess what seems like a hugely different sound to me ... still just sounds like me.
Jimmy J