They just put things back into the replicator without washing them. Otherwise you'd end up with as many cups as tribbles.

That was the issue with those travel mugs. Likely why they’re not sold anymore. Another 90s item that did not last.

The flared base is to keep it from tipping over during travel, which space stations are known to do a lot of.

No, these were for sale everywhere in the 90s, separate from the show. The idea was you could put your coffee on the dash or on a table in the back of an RV and not have it spill. There was typically a rubber, non-skid bottom.

Now, they’re nowhere.

NEXT question: is the base solid clay or is there hollow space between the cylinder and the corners?

I choose to believe it's for battle

Is that where Odo goes in, enjoys spending time as a coffee, and cleans it when he leaves?

The things a changeling will do for love...