A good chunk of the tone comes from his massive pick attack and use of roundwound strings. He sounded like JJ Burnell playing different basses in the '80s, even if the black P bass is the most iconic.
His tone was overdriven but not distorted. Really it nicely seasons the gristly, grizzly bits of a P bass tone. His first P bass was a '60s model with a rosewood neck and DIY green sunburst finish. It wasn't modded with more potent pickups or a high mass bridge. Find a bass with a '60s spec pickup, plug it into a tube head, 412 cab, and whack it hard! No need for Quarter Pounders or vivid distortion.
A few of his early bass lines were influenced by dub reggae, even if it isn't immediately obvious. Hugh Cornwell also had quite a thin, scratchy guitar tone, and the drums always sound a bit scooped on early Stranglers cuts. As such I think JJ has a unique tone because he had a lot of room to fill in the mix, and we get to hear a good chunk of his complete tone in the early tracks.
The other thing is that JJ is massively into martial arts. When I heard of this it seemed to make some of his playing and tone make even more sense. I guess his bass lines are sparring a little, and sort of muscly and punchy both in terms of tone and how they sit in the mix and align with the beat.
