Yes, you can add a roommate to your house — but only if you are the owner of that property.
It is not possible to add a roommate to an apartment.
The character you want to add must be standing next to you.
How to add a roommate:
When exiting your house, open the house menu
Select “Add Roommate”
Enter the ID of the character you want to add
Enter the duration of their stay
Pay the required fee
The selected character will receive a notification with a roommate request — they must accept it