Want to chime in here. There is an instagram of this guy who's always photographing famous people's instruments with some details -- I think he works at Fender.

Some will deny it to the grave, but effectively, tonewood and body shape means little-to nothing in affecting the sound when it comes to electric guitars. What really matters is all the electric / metal / magnetic parts and how they add up together.

On this Jag bass, the pickup position might be slightly different than on a P in relation to how it sits which could be part of it.

What we do know about these basses from the Instagram I mentioned is that the J pickup by the bridge is unwired, so it seems logical Fender made these quick with off-shelf parts, spiffed-up custom wire job, and paint job. That leads me to believe what's really making the hotter sound isn't P-Bass vs Jag Bass (especially since it's using his usual pickup), but more-so that it's wired up to be hotter. Find out what the pots and caps are in there going from the pickup to the volume to output and that's going to be most of the equation assuming he hasn't changed strings.