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The bed in the house is the exact same as BotW, and unlike other beds in the world, you can freely sleep in it. If we open up both botw and totk, the bed is the exact same one from botw. Meaning that at the absolute minimum, Zelda has been sleeping in Link's bed. However, if you go up to the bed, Link can still sleep in it. There are very few beds that are free in this game, and the only other beds you can sleep in is obviously communal beds, such as those at lookout landing for those who are sick, injured, etc. You cannot normally sleep in another person's bed.

2. Link's hair-tie, Zelda's diary and the Champion's photo.

If you have completed The Champion's Ballad DLC, and correctly transferred your save, then the photo will still be there. But more importantly, the house has a new addition. Behind the house is "Zelda's secret well" that functions as her new study. In that well, you will find one of the volumes of her diary, hot-footed frogs splashing around, and Link's hair-tie in the corner of the room. Link never took that hair-tie off in botw (only when wearing the ancient helm), so it's a bit suspicious that it's down here of all places.

Not only this, but if you read the volumes of her diary, both the one in the well and on the desk near the bed, there are a couple of things that stick out:

Firstly, Zelda had Bolson build the well because she needed somewhere to focus alone. If she was living by herself, this makes no sense (and, in japanese, rather than "my house" it's simply "the house" here).

Secondly, Zelda states that Link is always by her side. To the point his clothes have become worn. So as a gift she made him a new tunic (and she "can't wait to see the look on his face" when he gets it, just saying).

and third, she had to hide said gift in the castle, otherwise he would find it in the house.

3. The School and Symin

During the time-skip, Zelda founded a new school in Hateno, where she became a teacher. Due to this, a child who misses her visits the house at around 12-2pm and calls it "Ms Zelda's house".

However, if you talk to Symin, he has this to say:

So, clearly, this is not Link's first time here (which should be obvious, how exactly did Zelda get to Hateno and the house in the first place after botw's ending? Link brought her there). He even asks Link why he came back to the village alone without Zelda. Additionally, if you go upstairs and read the "School Notes" it says this:

It talks about the opening expedition under the castle happening soon, and reiterates Link being by Zelda's side. Interestingly, in that opening sequence Zelda says she is glad she didn't leave the Purah Pad behind. Meaning she's glad she didn't leave it at home.

But wait, there's more

There is a side-quest at the school where Link becomes a teacher / substitute teacher. When you finish the quest, one of the children says this:

They say they already thought of Link as a teacher figure even before this quest. To add onto this, there is another child who wants to become a hero and is "always on the lookout for baddies".

4. Manny

Manny is an interesting case. At first, like some other npcs, he seems to have been entirely memory-wiped to be approachable for brand-new players who didn't play botw, but then he has the following to say:

So, first of all, he doesn't call the house "Zelda's house" and instead calls it "the house the princess was staying at"

Secondly, he asks Link to "keep the naive, new-in-town act" to a minimum.

Third, Manny has another dialogue, he wants to "check every well" in town to find frogs for Ivee (because, similar to his quest with Prima in botw, he thinks that's what she likes). But after he says this, he seems to realize he should not have said that in front of Link, saying "Uh oh, no coming back from that one" in the smaller text as though he knows Zelda's well is off-limits. Combined with him calling the house "the house the princess was staying at" he seems to know something.

5. Two Place settings, etc

The house has two places set, two towels, etc. And just in general a lot of things for supposedly one person. There is also a bigger kitchen, and let's be honest with ourselves here, who is more about cooking and actually good at it, Zelda or Link?

Under the stairs in totk

VS botw:

Now, for this next section, I have to talk about things regarding another town in the game, a quest and a mechanic many have not unlocked yet. Continue to read at your own peril (I would hide it under another read more if I could)

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The Tarrey Town House

During TotK, after we wake up and leave the sky islands, we can complete a quest to unlock the ability to build a brand-new "Dream Home" in Akkala, close to Tarrey Town.

The quest involves baby-sitting Hudson and Rhondson's child, Mattison, who, if you'll recall, did not exist yet in botw. In totk, she is starting to gain independence and appears to be around 5 or 6 years old:

I doubt Link was sleeping rough for several years (and Zelda would never be okay with that). Anyway, Link babysits her, goes up into a hot-air balloon with the family, and then has to watch a tear-jerking scene between Mattison and her parents as she has come of the age young gerudo girls have to go to Gerudo Town and stay there until they get older and are ready to venture the world.

It is only after completing this quest, a quest completely about parenthood, that you are offered the "Dream Home" by Rhondson.

Additionally, an npc at the nearby stable has this to say:

So, the in-universe implications of the dream home is that it is a family home, something you only buy when you plan to marry someone or start a family.

Adding onto this, the dining table in the dream home has four seats instead of two. But there's also an interesting comparison to make with the Hateno house. The Hateno house had two seats and two places set, this house has four seats, but is clearly only being used by one person:

In other words, Link is living by himself in a house meant for a family. Which is depressing to think about, but in the context of the game makes sense.

But there's something else. In the new house pieces (including a study you cannot use yourself), all the photos on the wall are of Hateno Village:

Why would Link hang photos of Hateno in his new home, if it was not important or nostalgic to him? (and, as a minor detail, the same yellow flower is in both houses, presumably Link's favorite flower)

In conclusion, it's clear from all the evidence that not only were they living together in Hateno, but the brand-new house actually has it's own implications behind it (and further proves that Hateno was and is important to Link).