It's been quite a long time since I used a refillable mug at WDW but don't they have the mug wash thing? No mug wash on the ships.
As
@Cesar R M was explaining, germs are a really big deal on a ship. You don't see CMs stationed at the doors to restaurants in the parks & resorts handing antibacterial wipes to every incoming guest yet on the ships you do. They must be super vigilant on ships to prevent the spread of germs. The close quarters makes it absolutely necessary. On a ship it takes very very little for outbreaks to happen. Disney's commitment to their vigilance is a very good thing for everyone. Guests who are equally proactive and diligent really do help. It's the smallest thing, really, and awareness is key.
The paper cups at the drink stations are pretty small. They're almost like a kid size cup at fast food places. So far Disney has been cool about letting guests bring bigger cups to fill. We never sail without our Tervis tumblers. Using the paper cups to dispense adds a minute to the process of filling but its a small amount compared to the time lost if I get sick and get stuck in my stateroom. Or worse yet, quarantined to my room.
There is a medical clinic onboard staffed with a nurse and doctor. I've seen pics. It's like a mini emergency room. If you end up there it is not free if you require meds or treatment. If they think you could be contagious (as with stomach virus, for example( you will be quarantined to your room. When that happens the doctor says how long you must stay there. The crew will check in regular intervals to ensure you are there. They'll bring you food and other extras to make you comfortable but they will also be sure you stay quarantined. If for some reason you need care that cannot be given in the facility onboard the options are for you to transport off at the next port of call or, for emergencies that cannot wait, by helo evacuation. All the expenses for that will be yours. So, using the little paper cup, while annoying I know, isn't such a huge sacrifice to make when considering the big picture.