If it's a bare wire, then it's just to aid in preventing interference to your signal wires. It should be ground, but in your particular scenario, you can leave it free (or even better, cut it back and wrap the end of the larger insulation with electrician's tape.
What you're thinking the "main" power supply is, is actually your "Isolation Transformer". It took a while for me to understand what it actually does, but it's gonna convert your 120v electricity to something more monitor friendly. It'll also help prevent the monitor from shooting any bad electricity backwards into your cab, computer, wall, house, etc.
LEAVE IT IN! Also double check that the wires on it are going to the correct lugs (wall goes to 120v on one side, and your monitor should already be wired to the required lugs on the opposing side.
The blue box is a power supply for the arcade board. It's gonna take 120v in, and correct it to voltages your board requires. Much like a computer power supply, this power supply will provide +12v, +5v, -5v, and some grounds. It may also provide a number of other voltages, all indicated on the supply itself.
You'll be using a PC power supply so YANK THIS PUPPY OUT!
The chassis screw next to your "main power supply" (that we now know is called an Isolation Transformer (Because it isolates your monitor, while transforming voltages)) is actually a grounding screw. They decided to tie all the grounds (board, monitor, coin door, and probably a few other devices) so that there wasn't a floating ground, but one big centralized ground. This should also be tied in with your third prong on the AC plug that goes to your wall, to ensure your maximum safety.