You only rent a house. There is a limited number of addresses that are available in each zone, but an unlimited number of people that can dwell in that address.
Meaning, each house is an instanced zone, and once you begin renting the house, you gain access to your own version of it. Then you can begin placing furniture, books, quest items, pets, etc.
When you go to access a house at an address, you will be presented with a long list of players that have taken up residence there.