Coming from guitar nerd world, most everything with a DI option I've seen, also has a cab and/or speaker emulation choices, can loads IRs, etc. Even if the Guitarist is going to the PA, there is some sort of cab emulation happening, but in my bass preamp search, that seems to never be addressed. In practice, I've used a SansAmp, Ampeg DI and now a Battalion to a nice PA system & was happy. Never once did I think "Oh if only I sounded like an 810 Marshall cab" or whatever, but I'm still curious why it's not a thing for bass DI
It is a thing, but sound guys are used to taking care of it over the years... There's the OmniCabSim Deluxe. Mooer Radar And the Tech21 QStrip partly. Broughton Audio has some high pass and low pass filtering options, and a good fully parametric 3band EQ if you want to get really detailed.
Unlike the midrange of a nice electric guitar the most important part of a bass signal is not really affected too much by cabinet config.
I actually think it is a thing for bass and many of the pre-amps/DI's do in fact have some amp/cab sim aspects either baked in or user selectable/adjustable. The Tech 21 family of SansAmp products as an example certainly fall in this category. There are also a number of products that simply provide a pre-amp and EQ functionality with an emphasis on being otherwise transparent.
I agree that driver size and configuration are much less significant than many people give them credit for, but I would say that "overall cabinet voicing" varies considerably across and even within brands. I would also add that sealed cabs and ported cabs, independent of driver their size/configuration, tend to behave much differently IME, especially in the lower end of the frequency range. While it may not be as significant as in the guitar world, there are some signature amp/cab pairings that tend to be quite recognizable for the unique aspects of their voicing. From that perspective, I believe bass cabinet design can be a very significant in how a bass sounds to the audience. As always, YMMV!
I used to have a V1 Sansamp but got rid of it because of the lack of mid controls. I think I read somewhere they were based on an SVT tone, but have never seen anything regarding it that mentions cab or speakers (at least with that old one- haven't looked at the latest and greatest from Tech 21 yet)
They've got a new BDDI (V2) with a selectable mids control, selectable bass control, and a better (more usable) presence control taper. There is also a VT Bass DI (guess what it's based on) as well as Character Series pedals that work well for bass and guitar. Essentially the the SansAmp concept is that you're sending a signal that sounds and feels like its the product of an amp and cab. The user manuals on their website have a wealth of information and the people at Tech 21 have always been happy to educate me when I called with questions. There are a lot of other great companies doing similar things as well, but I focused on Tech 21 as you had mentioned their product and they are admittedly a favorite company of mine.