Doug was always big into the Line 6 stuff, as well as what Voodoo Labs was making, especially the analog chorus pedal. He changes pedals and gear around all the time, and if you ask him about what he used at some point in time in the past, he probably won't remember.
In this live clip, he's playing through a monster rig, like he always does, and has PA support. I've heard him get that sound in small bars using a little combo, so the big rig isn't a key thing. For more down-to-earth attempts at this chorus sound, I'd look into a Boss bass chorus pedal, which processes the high end but leaves the low end alone, so you don't lose it. This way, you get the chorus sound on the highs while the low end still happens, which is what I hear when Doug plays.
A Spector bass, played with the "flamenco slap" tecnique you see him doing here, is a big help in getting that sort of tone. DW usually wasn't stingy with a bass and treble boost from the onboard preamp, either. Add that up with a good chorus, and you'll be close to the sound you're hearing on the clip. My computer speakers have no bass in 'em at all, but it sounds like he might be using a Whammy pedal set to tune the sound down in pitch, which might explain what you hear before the first verse. He isn't sliding his hand down the neck there, so the Whammy is a good guess at how he's doing that.
Doug's from the Hartford, CT area. Years ago, all of us CT players that looked up to Doug always heard him say something like "All these boxes and pedals are cool, but for the most part,
I'M the effect." I think he's right, because Doug sounds like Doug, even without any effects.