What is a “domain game”?

In one word: politics.

More words from here:

Domain-level play is an old term for giving RPG characters political power in the form of ”domains” they control, be those guilds, corporations, or part or all of a sovereign nation. Historically, Dungeons and Dragons campaigns would eventually see the characters having enough wealth and influence to purchase a stronghold, which would give them not only a base of operations but also a parcel of land to see over. Once the characters were officially nobles in this way, a whole new area of storylines could open up, involving courtly intrigue as well as broader politics. At a default level this would insert characters onto a bigger political stage, but it was both possible and for some palatable to start changing the course of politics in the world in which the game took place.