Unforatunately, you can only use your mug at the resort hotel where it was purchased, not "anywhere on WDW property."
We don't buy one because we're in the parks so much - we only really eat breakfast at the resort restaurant (and sometimes just have a quick breakfast in our room instead!) and no other meals at the hotel, so it doesn't make sense to get the mugs. If you're at your resort a lot, then it makes a lot more sense.
Plus, if you just use it in the morning, you have to either lug it around with you all day or bring it back to your room before you go to the parks. We tried it one year at POR, but it seemed like a hassle to bring it back to the room every time.
(By the way, some people seem to be assuming the OP was trying to get away with something -- it seems to me they are only asking a question to clarify the rules. When you hear "unlimited refills," you could easily assume that it's anywhere on WDW property.)