The "guitar" appears (visually and aurally) to consist of four octave courses, so it's like a mandolin that way but constructed like a guitar. Mando-guitar, perhaps? I don't hear too much prog influence (so, you can sleep tonight) other than perhaps the style of the 'cello and violin playing (derived in small part from Ponty's work with Zappa in '73?). There's a bit of a New Grass/Flecktones funky edge to it. I'd call it Aggressive Americana, you know, so you know which section of the record store to go to. It is good; no disagreement there.
I did a little Googling, like all the cool kids, and there are indeed 8 string tenor guitars that look just like what she's playing, but it turns out it's an "octave mandolin," so you were in the ballpark. Here's the page for the builder of the very instrument in the video.
At $5,800 it's not exactly something for amateurs or dabblers. ;)
...and now everyone but you and I are fast asleep...