As an owner of a Squier Bass VI, I have to say that this particular sound is due to the unique facets of that instrument, in addition to her using what looks like a Vox AC15 guitar amp, an open back combo. Short scale, 6 strings crowded onto not that wide of a neck(a gauge thicker than guitar, of course, but not quite like what you find on a 4-string), three single coil pickups that are essentially trickle downs from a Jaguar guitar(somewhat like a Strat). Which pickups are on have a lot to do with it as well; the middle one is reverse polarity, so interacting with either of the others(or all three on)will provide a hum cancelling sound, but the front and back only is something entirely different. The other switch is a bass roll-off. The amp used, if indeed that's the one on the recording, is not ideal for low frequencies, but then Carol Kaye used an open back Fender Concert guitar amp on some of the most famous hit songs ever.
You have to consider the history of the instrument…it was designed to go up against the Danelectro 6-string bass/baritone guitar used in the 'Tic-Tac' style on certain country records of the late 50's/early 60's, and also as an instrument that guitar players could easily transition to bass with. But that doesn't mean you can't get fat, bassy sounds out of it. Jack Bruce did just that on the first Cream album(many mistakenly think it's the famous Gibson EB-3), but I think a good portion of the sound is due to studio compression and his use of a Marshall amp. As well, one time Soft Machine bassist Roy Babbington got a similar tone on a VI. It's just that you really have to amend your technique to play it.
I've checked that band out myself, they have an early 80's dream pop/goth(4AD/Factory labels)thing going on, but then the Cocteau Twins(and Cure)employed a VI back in the day, so it fits.
