What is the name of this Chinese Fish Soup where guests cook their own fish in broth?
Sounds like hot pot.

There are a zillion variations, with different kinds of broths and different things to cook in it, so I don't think there's any one recipe name you could search for to duplicate exactly what your friend had. It's possible that knowing the region he was in would allow some informed guesses from folks with some local knowledge (not me). Note that it's common to cook a lot of different things in the broth all at the same time, not just one thing per meal.

So if you want to try to make something similar without a lot of work, I suspect you'd just want to a broth you like (or a soup base from a Chinese grocery store - hot pot is popular!) and whatever simple fish sounds good to you. If you want more elaborate things, try Google; there's too much variety for us to try to cover it here.