The ones in DS9 were custom-made, but were based on the Highwave HotJo Travel Mug design. It's also my preferred tea mug, but I didn't even make the conscious association with DS9 until years later.
They used to make them in black, which is the one I had until I dropped it. I had to get a white one, which was the only one they made for a while, then they made a few in colors, but not black, which is what they're selling now. My wife got me the purple one for Christmas, but I finally found a black one to replace my original on eBay and I drink tea from it every day.
(To be fair, there were plenty of black ones on eBay before that, they just all had stupid logos on them.)
https://highwave.com/products/hotjo
Edit: Wow, they're already sold out of the purple. They must make them in really small batches.
Mine is something called a "Mobile Mug". It's a hollow plastic cup (polypropylene appparently) with a rubber friction-fit bottom. I'm just barely old enough that as a young law student, mugs of brewed coffee were still the most common legal stimulant for 1L's in the library. I lost the lid years ago, but in honor of getting back to my DS9 re-watch mixed with new-watch, I dug it out for some French press rocket fuel.
Then I promptly forgot it was on the table and poured the coffee into a random Starbucks-knockoff plastic to-go cup.
We had a plastic one when I was a kid too and I loved it. That's why I eventually discovered the HotJo one. You can see from the stills here that the ones in DS9 are slightly different (I wouldn't be surprised if Highwave themselves did the custom job), but clearly based on the HotJo mug.
I'm sure Mister King is problematic in many ways, but for a Southern corporate lawyer from the early 1900s, he's pretty boring to judge from a quick internet search.
