This is pretty much exactly the same as achievement "Fur-iendship Bracelet"
Simply have your friend step out of the play area and move in yourself, hold your right hand up toward the top right of the screen to swap players, you can then sign in and you will take over from them (it still counts as their game.)
Then open the toy box > map > house > shop, and simply buy any piece of furniture, once it's bought Achievement Unlocked.