What should I do with the ground wire on a replacement bathroom fan motor?
As long as the motor is solidly mounted to a metal bracket and in turn solidly screwed to the fan housing then the ground wire may not be needed. The old fan may have gotten it's equipment ground from this method so the new one can also.

It also possible that this replacement motor is also sold for appliances that have plastic housings so the ground tail is required.

If there is no grounding means existing then the same really applies. There is nothing I can think of that says if you have to replace a small motor like this that you have to upgrade the circuit feeding it.