See: https://en.wikipedia.org/wiki/Board_game_caf%C3%A9

How would you tag a board game cafe? The main attraction is the ability to ‘rent’ a table and play a variety of board games offered by the establishment. Oftentimes these places will sell cafe-type drinks and sometimes alcoholic drinks and food.

In my opinion, “amenity=cafe” and “amenity=pub” are not sufficiently descriptive because the implication is that the primary use is a cafe or pub, respectively. Adding a tag like sport=board_games or board_game=yes, while helpful, only indicate the availability of board games for play at a venue, but do not address the underlying issue that the primary use of the establishment is to play games not consume coffee or libations.