How to cut edge of vanity
Yes, you can cut it. Use an angle grinder with a masonry blade. I have seen it cut with a jigsaw as well, and a circular saw for flat pieces, but I prefer the angle grinder for this type of job. Cut it slightly long, then use a belt sander to take it down to your exact line and smooth the cut edge. This will (obviously) be messy, so do it outdoors. The "cultured marble" material is cast resin and will give off a fiberglass reek and quite a bit of resin dust, so wear goggles and a particle mask or respirator.

Cutting that much off the side will offset your sink basin from the plumbing, be prepared to change the drain and supply-line configurations. Cut it so that there will be a small gap (1/8"-1/4") left on the left side, as the walls are never perfectly square. You will then use caulking to fill that gap as well as where the backsplash meets the wall.

Another option would be to screw a strip of wood to the left end of the cabinet and to the wall on the left ("nailers"), then cut a piece of oak plywood to cover and fill the gap, leaving an unused hollow space behind. Stain and varnish to match the vanity.