Cubby house:
means the same as the previous 2.
In Australia cubby houses (or colloquially "cubby" or "cubbies") were historically built by children (with perhaps some help from older persons) from found or scrap materials often in out-of-the-way places in the garden or in wasteground areas. They were by their nature impermanent structures often lasting for a season or two or just for a school holiday. Latterly however, with safety concerns becoming more of a focus and children playing less frequently outside of the family home's garden, construction has been more formalised with kits and even fully-built units being available commercially
game room:
A recreation room, especially a room used for table games, such as pool or table tennis.
It is used for a variety of purposes, such as parties, games and other everyday or casual use. The term is common in the United States, but it is not used in the United Kingdom. Often children or teenagers will entertain their friends in it .