Interesting song, I haven't heard Eilish before. I'm a hobbyist house producer that does some mastering on the side, so I can say what some of that "weirdness" in terms of the volume is doing from the engineering perspective.
It's called ducking or sidechain (ducking being the more correct term, but sidechain being more common). It seems around 20 years ago, producers realized that their kick could sound more powerful if they had everything else around it fall away when it was playing. To do this, you use the sidechain input of a compressor (the Alesis 3630 being a favorite for this) and run the kick into it to turn the main input sound of the compressor (everything that isn't the kick) down only while the kick plays. For a clear example of what this sounds like in house music, see "I Remember" by Deadmau5. In house, we realized that this added the the feel of rhythm in our music - not only is the main driving beat of the song strong as ever, but everything else is reacting to it in the same rhythm.
Modern production has taken this a step further and now uses "ghost kicks" or just straight volume automation to achieve the same effect (the ghost kick just being a kick that you send into the compressor to trigger it, but you don't actually hear the kick). Either could be used to get the effect in this track. If you think about it, there's no drums at all for the first 30 seconds, yet you have this volume automation giving you a rhythm to latch onto. Sidechain these days can also be used selectively, so as you noted it brings the vocal even more forward because everything else is being ducked in volume to the rhythm of the song.
It sounds to me like the effect is even more apparent because, most likely, there's some saturation or distortion on the sounds being sidechained after their volume is being automated, which would mean that the distortion is also pumping in rhythm with the song.