Funny you ask! While not specifically answering your question (though I will, at the end), I am very specific about my coffee mug(s)! On my first return trip to Ireland I picked up a mug in the airport - green with a little white sheep on the side, one on the inside, and one on the bottom. I started using it as soon as I got home, even taking it to the university where I worked and using it there every day. It lived on my desk and reminded me of that amazing trip. Even still today, more than 5 years later, that is MY mug, and I use it almost every day. I always joke with Jean that I can't live without that mug, so if/when it breaks, I'm booking a flight back to Ireland right away to get a replacement! Oddly enough, by pure coincidence, my second and third choice mugs also both have sheep on them (one is a Northern Ireland/Giant's Causeway mug and one is a knitting mug).
Anyway, to answer your question, I put Jean's coffee in a reusable plastic mug, the kind with the screw on lid and just a hole to drink out of. We specifically bought the kind that doesn't have any kind of snap or slide closure over the drinking hole, since I've found that it's just more stuff for coffee and germs to get stuck in. All of this said, given the hole and the plastic material, they probably don't keep it hot for very long, though Jean guzzles coffee like there's no tomorrow, so who would know?! If I were packing him coffee that I knew would sit for a while or that I wanted to keep hot, I would pack it in a metal thermos that had been primed with hot water.
Hope this helps! Curious to see what everyone else uses.
Mandy