move is the verb of the sentence. Buying is the imperfect adjective which describes what you do while you move around the board. The comma(,) shows that this is added information. If you didn't have a comma(,), but you had the connector "and", then you would say buy. Example: In Monopoly, you MOVE around the board and BUY houses and hotels. (Now the 'buy' is not a description of an action, it is the action.)

It is due to the sentence structure. If you used others words to connect the sentences then buying could be used> "In Monopoly, you MOVE around the board AND BUY houses and hotels." You could also change the sentence around> In Monopoly you BUY houses and hotels while MOVING around the board. In Monopoly when MOVING around the board you can also BUY houses and hotels if you own the property. In your first example, changing the BUY to BUYING does not work and is not easy to say.

Buying is extended in duration, repeated. But, "you moving" is not correct. If you like In monopoly, you keep moving around the board, buying ....