If the player avatar owns just a room (like the Hunter's bedroom in Metal Saga, which can be redecorated, or Shepard's cabin in the Mass Effect games) does that count? What if they lead a faction which, in turn, owns a building, like the castles in Suikoden games? What about areas that they unquestionably own, but that doesn't quite contain a house for them to live in, like the Geonet in White Knight Chronicles?

What if the player is a turtle or hermit crab? Does that count as well?

Minecraft.

Done.

@Ravenlight: Only if the avatar can suck themselves into the shell and walk around it. Considering how often this happens in cartoons there has to be at least one game that has this.