I have an older parts Jazz bass with a Warmoth neck (maple/rosewood/steel reinforcing rods), ESP body that might be alder though I've never ID'ed it, Badass II bridge, and DiMarzio Ultra J pickups in a passive layout. This bass is too heavy for me to wear on a strap all night (10.5 lbs.), but I absolutely adore its slap tone and I attribute much of this sound to the DiMarzios.
For me to get close to Ida's tone here, I'd use this bass, but I don't know whether a specific combo of woods is essential for it.
I swapped a set of DiMarzio's Area J's into that bass to try something different, but I didn't care for that layout. It just didn't have that big 'n snappy sound that just begs for slapping. I also tried the Area J's in another bass with a preamp and that seemed to work a bit better. I got a different sort of useful Jazz sound (typically described as more "mid-forward"), but not so slappy like the Ida tone.
I'll also try DiMarzio's Model J's pretty soon - those are rumored to be slightly less scooped than the Ultras while still having lots of the warmth and fatness that make the Ultra J's attractive for me. The Ultra J's (and apparently the Model J's, too) also give me a nice useful tone with the neck pickup solo'ed. Not a clone of a P bass, but more wooly and "honky" compared with the scoopy sound of both pickups full open on a Jazz.
When Ida is slapping through that first 0:55 or so, it seems like her sound also has some overdrive going on, but I'm not terribly experienced with bass OD. Not sure if she has some drive included in that amp or maybe there's a pedal we can't see.
That second section after about 0:55 sounds like either a solo'ed bridge pickup or the blend knob turned at least partially toward the bridge, plus a little fattening up from a preamp either onboard or at the eq on the head. Probably hard to get that without a MM pickup at the bridge, but my DiMarzios are humbucking J's that might get near that sound with some eq help.
