Bottom line up front: since one of your concerns is "Better performance in low light (photos get noisy around ISO 800 on my 600D which is frustrating)", the 6D has around a stop better high ISO performance than the 7D II (which in turn has only a marginal half stop or so advantage over your current 600D).
That said, since you "shoot mainly portrait and fashion photography", you'd probably do better to invest in lights/flashes. During your studies, I'm sure you came across the phrase "photography is painting with light". On top of which, as I'm sure you also know, as ISO increases, dynamic range decreases. So (in controlled situations like portrait and fashion) it's always better to add light(s) than increase ISO.
As far as focus points, while the 6D doesn't have the 60+ focus points of the 7D II, it will still be an upgrade from the 600D. (Especially since in portrait/fashion, it's common to shoot in, well, "portrait" orientation, which is why it's called that.) On top of which, in controlled situations, it's always better to use just the center point and reframe (so the camera doesn't "guess wrong").
And sharper images generally have more to do with the lens than the camera (body).
I studied at a photography school and graduated at the end of last year and I hope to make a career out of photography, so I really want to upgrade from my Canon 600D. I'm saving up for a new camera, but I'm not sure which one to choose. I've been researching a lot and it sounds like the 5D Mark II (or III?) is the best choice, but that's out of my price range. Would the 7D Mark II be a good choice? Or should I rather upgrade to a full frame camera like the 6D?
I shoot mainly portrait and fashion photography. The things I'm looking for in a new camera are:
- Better performance in low light (photos get noisy around ISO 800 on my 600D which is frustrating)
-More focus points (my camera only has 9 focus points)
Any help would be appreciated! I'm struggling to make a decision.