You can fill your mugs up at park *water fountains* if you like. Just no sodas, etc. I would be mortified filling up a mug at Ice Station Cool. That's tackier than flypaper!
Anyway . . .
Animal Kingdom and the water parks sell their own larger type refillable bottles/mugs that you can use for the DAY you purchase them at THAT park only.
MK, Studios and Epcot currently do not offer anything like that.
Resort mugs are designed to be used at the resort in which they are purchased- typically for sodas, iced tea and coffee.
It's a good deal if you will be eating meals at your resort. Just sleeping there, might not really save any money, esp if your resort doesn't have 24 hour access.