They kind of had the same moment in TFA when Chewie gave her that look as she sat down. It's obvious that Chewie wants her to have the Falcon... he's just going to be her side-kick now instead. She may be Luke's daughter, but she's also Han's Niece.
Strange, just checked the wikia and it says "under chewbacca" http://starwars.wikia.com/wiki/Millennium_Falcon
I think that given that the Falcon was lying abandoned on Jakku, sitting in an apparent state of disrepair that not even Rey who seems to have some skill at recognizing operational and/or functional equipment didn't want to use it when she was running for her life invalidates all previous claims to ownership including Han and Chewie. As such, Rey and Finn can claim salvage rights when they make use of the Falcon to escape. The fact that when Rey recognize Han's name when he and Chewie show up would suggest she cedes any rights to the ship she might have back to Han and Chewie. As neither she nor Finn explicitly acknowledge Han as the sole owner of the Falcon, it can be implied that Rey is ceding her rights to the pair of them. When Han dies, this leaves Chewie as effective sole owner of the Falcon. Now this is where things get tricky. IF Rey is Luke's daughter, I'm sure Leia will pass whatever rights to the Falcon that she controls over to Rey without any objection from Chewie. But if Rey is just some random orphan, in the words of C3P0, Let the Wookie win.
It should go to Kylo. He can paint it black and slay all of your childhood heroes while piloting your favorite vehicle. Just kidding, it should go to Chewie and Rey.
Whoever it is he has to be comfortable with them for sure. Rey is an obvious choice due to the fact that Han trusted her but I just think she'll be busy. I do think it'd be awesome if she was a lightsaber-wielding pilot, much like Luke in ANH in this case much more powerful though. And ownership in GFFA is very fluid it seems. Like someone mentioned R2-D2 and C-3PO went from a variety of owners, in the former's case from Padme, to Anakin right?, to Bail, Leia, and then Luke. And that was never done officially. Also how long was the Falcon on Jakku too, do we know?
"You shall receive one quarter portion." Reminds me of a scene in Dickens' Oliver Twist or something.
Episode IX-Kylo kills Luke, Leia and Chewie and then blows up the Falcon with 3PO and R2 on board. Iris out, saga ends.
Chewie, I hope, I want more aliens too have clear agency in star wars. I would also prefer him to be the pilot and Rey to be a co pilot.