When we were at Blizzard Beach recently, the CMs specifically said that a person could buy a refillable mug and then the whole family could share it. So that's not only allowed, but was actually encouraged.
I've always shared resort mugs with my kids. I would have coffee in the morning (assuming I didn't bring my little coffee maker, which I've never hidden after use), the kids would have soda in the afternoon and if it was cold I might have hot tea later or they might have hot cocoa. As to reusing the mug on later visits to the same resort... Never would I admit to such a thing.