Well, I think she is clearly set up to be a Skywalker. Being Luke or Leia's daughter would set that up fine for me - I'm not sure Star Wars carries the same conventional married last name traditions as people in the real world.
That said, the reactions by Han and Leia were pretty cold if she's a daughter or a niece, assuming they know who she is as well. They have been searching for Skywalker all this time - would be kind of odd to treat his daughter as something of a stranger if they really knew who she was. Recall that Vader locked up his daughter on the Death Star and tortured her without realizing she was his child, so wouldn't be crazy if Han or Leia didn't really know Rey...especially since "Rey" might not be her given name.
Nick - was the book worth reading? Curious to know more about how Ren knows of Rey prior to the events in the movie.
Regardless of how the marital naming goes in the Star Wars universe, Leia has never taken the name Skywalker and is General Organa in the movie...not even Organa-Solo. The book makes it seem like they were married. The movie didn't quite come out and say that. Heck, we aren't sure if it is Ben Solo, Ben Organa, or Ben Organa-Solo. Maybe the "I used to be" line could mean he took Leia's name and became Han Organa, but I think it was just meant to be a funny line.
That all being said, if Rey is Han's or Leia's or both of their daughter, then it is really cold that they don't say anything. I think she gets much more from Han than Leia as he seems a bit protective over her with the job offer and all. If Leia could feel Han's death, I would think she could sense if Rey was her daughter. Coupled with the fact that if they had sent their daughter into hiding when Ben became Kylo Ren, I would think it wouldn't be that hard to put two and two together even if they had no clue where their daughter was sent. In conclusion, if Rey were a Solo or Organa, while still a grandchild of Anakin Skywalker, she wouldn't directly have the Skywalker name.
Now, if Han and Leia had no clue Luke had a daughter, that could explain things. If they knew that he did, but never met her since she seems to be three or four in the flashback, it is understandable that they don't recognize her off the bat, especially if as you said, Rey is not her true name. I believe that Kylo Ren's age has been worked out to be 29 in this movie or just shy of 30. Pablo Hidalgo states he was born just after Endor, so I would say 29. Rey is 19. If she is abandoned at age 4, assuming it is due to Kylo Ren, that would mean he has been with Snoke for 15 years. Not sure if they mean for Rey to be older in the vision, if Kylo Ren first went to Snoke leading Rey to be abandoned then Kylo Ren came back to take out the students at a later date, or something else. Since Rey appears quite young in the vision, I would think it would be easy for Han and Leia either to not know she exists or who she is if she is Luke's daughter.
My take is that Han and Leia may have been aware of Luke having a child and I think that they work it out during the movie. I believe that is why Leia sends Rey to find Luke and why Chewie and R2 stay with the ship. I don't think it would make sense for Leia to send Rey to Luke for Luke to tell her that Han and Leia are her parents. It would also make sense if Luke was the one who left her on Jakku if Rey just happens to not be related to the Skywalkers at all. It doesn't make sense for her to be sent to find Luke if she is Han or Leia's daughter.