I've got several of those monitors with that same flyback.
Yes, that black wire needs to be grounded to the frame. It's bleeds off high voltage and if left dangling while the monitor is on will have the same HV potential as the second anode!!
As a matter of fact, if you run the monitor in a dark or dimly lit room you would see a high voltage corona around the end of the wire (but I don't suggest you try it).
The plastic box itself should also be mounted to the frame, it should have a couple of mounting tabs for that purpose. The mounting doesn't provide any electrical connections, so if the tabs are broken you can just secure the box with a wire tie.
D