The monitor should be floating, the outside of the Net City should be grounded. Any connected hardware you can choose to ground as you see fit.
So why wouldn't you ground everything? It's because often the conductive hardware enclosure is also tied to common (DC) ground. I don't own a MiSTer, but my guess is when you tie the MiSTer case to Earth ground, you also tie common ground to Earth ground. This may or may not result in interference.
Normally arcade hardware is left ungrounded. The PCB tray is made out of wood for a reason.
However, there's no harm in experimenting. If you feel better that the metal case is at ground potential, then go for it.