MCTS can be applied to imperfect information games, in fact it is quite robust against uncertainty.
It's AlphaZero's deep neural net component, that's used to learn an evaluation function and move orderings that will need a substantial redesign to take into account imperfect information. The difficulty of this redesign will vary considerably between games- in some games, information is gained throughout the game, by observing another player's moves (e.g. in Poker), in some others an initial state (e.g. a starting deal in card games) dominates the probability that a certain possible board state is the real board state (e.g. Bridge) [1].
On top of that, AphaZero's deep net has the shape of the board and the legal movements of pieces on it hard-coded, as part of the net's structure. That would also need a substantial redesign to accommodate a card game, or any other kind of game without a board and without pieces that move on it. In fact, different card games will require different architectures, most likely. It's very hard to see how, e.g., the same neural net structure could be used to encode both Bridge and Poker rules - and still allow learning chess, shogi and Go.
Given the great variety of board games out there (and that's only classical games, I'm not even considering modern board games, like Settlers, etc) a lot of very hard work would be required to even train AlphaZero to play any game that's not very similar to chess, shogi and go. Not to mention, training AlphaZero is very expensive (wikipedia quotes a cost of $25 million for AlphaGoZero, AlphaZero's predeecssor, and that's just to buy the hardware [2]). So I don't see how or when they'll demonstrate the "general" game playing power of their system.
Basically, I think all that stuff about "generalized" game playing is just so much pointless bragging. The way DeepMind designed AlphaZero is exactly how everyone else has designed their systems- hard-coded with structures appropriate to the targeted game (e.g. boards and pieces, etc). DeepMind are clever in that they chose three very similar games, and then threw an immense amount of money on the problem of solving them all in tandem. And still they had to train different models for each game. That's just no way to get to general game playing.
___________
[1] See chapter 5. Adversarial Search in AI: A modern approach, 3d Ed. for a discussion of imperfect information and stochastic games and the difficulties of designing evaluation functions for them.
[2] https://en.wikipedia.org/wiki/AlphaGo_Zero#Hardware_cost