When DS & DD were younger, we didn't purchase the refillable mugs for them; just got 2 of them. Airports weren't as strict, so we packed a bunch of juiceboxes (applejuice) and gave them those. We bought a larger milk at the gift shop for breakfast.
Last time there, we were on dining plan so just got them for free. They'd usually get lemonade, punch, poweraid or the occasional soda, as well as get bottled juice or milk with their meals.
It's a lot cheaper for Disney to give refills on sodas/lemonade/punch than on juice or milk. I don't know of any restaurant locally that would give free refills on those, so didn't really expect it.