Depends on the motherboard really. I have one board where, despite metal standoffs, screws and back plate, ceramic discs are inserted between screw/standoff to isolate the board. Not sure why this but it is an older board (300MHz era) so it doesn't concern me much anymore.
Back to the question at hand:
My question is whether this connection is necessary. If i mount the board on the plastic standoffs and theres no longer that connection between the motherboard (through the standoffs and case) to the psu, will i run into any electrical problems?
Depends on your luck. I run motherboards without cases at all for long periods and never run into a single problem. I have a few boards where ground was broken for some reason and I got all sorts of really weird behavior that were almost impossible to track down. Weird memory errors, odd HDD reads, graphic glitches, random power switches (a convoluted problem involving a refrigerator, dirty power and a broken ground).
I'm not really an expert at this but from what I understand, the whole point of having those ground contacts on the board like that is to create as wide a path for ground to take. When you restrict or break the path to ground then you create an area of unknown potential or capacitance(?). On older arcade boards, I guess this isn't much of an issue. As the board speeds go up, having a well established ground becomes more important.
I say this. Try it on wood. If you start getting unexplainable errors or odd behavior, then buy a sheet of sized conducting metal, drill your holes (or find an old or cheap compatible PC case and cut the mounting plate out) and remount inside the cabinet using metal standoffs.
I honestly don't know if I would go with what swamprat96 suggests using braid. I think the potential for the ground braid moving and shorting out other components or coming loose and shorting things out makes me too uncomfortable to use as an option. Maybe use insulate wire and crimp/weld
using ring terminals might be a safer option?