Out of curiosity: would it be possible to train a neural network (like KataGo) to get good at several games? I'm thinking Go and Hex, for example, or Go and Keil. Would such a net learn to adapt some of the concepts it learned from the first game to the second one?

It's certainly possible in theory. You just force your two networks for the two games to share some of their structure. I don't think it would work well in practice, though, for two main reasons:

1) All the AlphaZero-type networks have a structure that is explicitly related to the geometry of the particular game. So for example the AlphaZero network layers are all 19x19. If you wanted to make a common structure you'd have to make it much harder to let the network reason about specific coordinates on the board (or you'd have to keep so much of the game-specific structure that adding the common stuff wouldn't really have an effect).

2) In my opinion, almost all of the common "how to reason about games in general" features reside in the MCTS part of the algorithm, which is already the same from game to game (excepting for the setting of some hyperparameters).