In Tears of the Kingdom, Clavia, the wife of the mayor of hateno village, takes care of Zeldas house while Zelda is gone. Clavia sweeps the floor in the house and keeps it tidy. As we know, Zelda has been living in Hateno village, the now largest hylian settlement in Hyrule, founding a school and studying in her secret well. Everyone in Hateno village knows her, she's the princess of course they know her.
If you approach Clavia while she's on her way home from cleaning the princess' house, Clavia will tell you something along the line of, she's tired and heading home, and if you need somewhere to sleep you should check in at the inn.
Zelda has most likely been working with the mayor, and knows Clavia and that's why Clavia cleans the princess' house. There is no fucking way that IF Link lived together with Zelda, she wouldn't know.
So no, Zelda and Link do not live together in Hateno village, and if you talk to the villagers it doesn't seem like Link even visits Hateno that often. Literally the Zora in Breath of the wild recognize Link better after 100 years than the villagers in Hateno did after 6ish years. Most likely Link lived and worked at Lookout Landing and he gave away his house to the princess.
I'm not saying Link and Zelda aren't a couple, they do seem to be very close, but they definitely just did not live together.
I don't want to start a completely new game just to prove that Clavia says that you should check in at the hotel, but here is a screenshot of what she says if you talk to her a second time while she's on her way home. Like she really doesn't want you to sleep in Zeldas house.
