How to Align vanity flush without creating a gap from baseboard?
You either cut the back bottom of the vanity to clear the baseboard, or the baseboard to clear the vanity - in this case, with &quot;legs&quot; rather than cabinet sides, (so small rather than large support for the base) cutting the baseboard seems like the obvious choice. With plywood cabinets that reach the floor and sit on it full-length, coping out for the baseboard would have no significant effect on the structure.
Third possibility would be to return the vanity and get one with legs positioned differently, but that may not be practical.
Either the wall or the back of the vanity is not plumb, evidently, based on the first picture with no baseboard.