Explanation for Kongou's joke:

The Yamato, while prized as one of the most powerful battleships of her time, was ridiculed as being a mere "hotel" for her crew, as she scarcely participated in combat. The KanColle Yamato presumably takes great offense to this nickname, even moreso when Kongou attempts to make a lame joke about it.

Kongou's joke is about how coffee/milk = western hotel, while Japanese tea = Japanese inn.