When asked who she is by the old woman on Tatooine, the former Palpatine declares herself Rey Skywalker, and she becomes a new part of the family. So, as far as the canon goes, yes, Rey is considered a Skywalker.
Later, Star Wars: The Rise of Skywalker revealed that her grandfather was actually Emperor Palpatine (Ian McDiarmid). It was also revealed that Palpatine was alive, and went up against Rey at the end of the film. In the interview, Ridley talked about her view on Rey being revealed to be a Palpatine.
Rey is not a Skywalker by blood, as she is not related to the Skywalker family. However at the end of The Rise of Skywalker Luke and Princess Leia both appear to Rey as Force ghosts, and Rey claims the name Skywalker symbolically.
Rey is not a Skywalker. She is technically a Palpatine.
Following Palpatine's final defeat and her own resurrection, Rey adopts the name Rey Skywalker to honor her mentors and their family legacy and renounce her lineage. As the last remaining Jedi, she makes it her mission to rebuild the Jedi Order.
What is the connection between Kylo Ren and Rey? As Kylo Ren himself says in The Rise of Skywalker, he and Rey are bonded not just by their familial legacies (Ren as the grandson of Darth Vader and Rey as the granddaughter of Emperor Palpatine) but by their unique connection in the Force.
She does not have a baby bump, her hand never goes to her stomach to indicate she's carrying a child, and there wasn't exactly any time for her to have conceived a child with anyone in the course of the film, including Kylo Ren/Ben Solo.
Rey is the first woman whose journey to becoming a Jedi has been central to a “Star Wars” trilogy. But our data shows she's played a smaller part in her story than Luke and Anakin Skywalker did in theirs.
Move aside, Anakin Skywalker. Master Yoda has risen above all others to claim the top spot as the most powerful Jedi of all time and the true chosen one according to IGN's audience. He won pretty handily as well, as he was victorious in 89,756 of his 95,243 battles and had a win percentage of 94.2%.
By the end of The Rise of Skywalker, Rey had become the last known Jedi of the Skywalker Saga. With all that being said, Rey was not left as the last remaining Force-sensitive character.
That's ok, but a little disappointing. Since The Last Jedi outright says Rey's parent were not important (from a certain point of view), and Luke shows no indication that he knows who Rey is, or even cares, it'd be really hard to justify that original idea.
Rey was born on Hyperkarn in 15 ABY during the rise of the New Republic. Her father, Dathan, was a bioengineered Strand-Cast cloned from the genetic template of Darth Sidious, the Sith Lord who ruled the Galactic Empire as Emperor Palpatine.
A crystal embodies the Jedi as the heart of the lightsaber, and pure kyber has no color until it's bonded to the Force user. This process likely resulted in Rey Skywalker's lightsaber being yellow.
Born in 15 ABY on the planet Hyperkarn, Rey was the daughter of a young man and woman, whose names were lost to history. Her father, Dathan, was the cloned "son," a bioengineered strandcast, of the Sith Lord Darth Sidious, making her the granddaughter of the fallen Galactic Emperor and Dark Lord of the Sith.
Anakin Skywalker is the son of Shmi Skywalker, born without a father through the Force. He is the secret husband of Padmé Amidala, the father of Luke Skywalker and Leia Organa, the father-in-law of Han Solo, and the maternal grandfather of Ben Solo.
One could theorize that he bore a child through some sort of space artificial insemination, or something closer to what we've seen in canon Star Wars material. In the canon comic Darth Vader No. 25, it is revealed that Palpatine somehow used the Force to manipulate midichlorions to impregnate Shmi Skywalker.
He died of old age on Dagobah. Fans of Jedi Master Yoda likely know that he retains much of his power even after passing into the Living Force.
Obi-Wan Kenobi, during his confrontation with Vader on Mustafar, stated that he believed Anakin was indeed the Chosen One, but that his destiny had been altered, and the prophecy left unfulfilled. Anakin fulfills the prophecy by killing Palpatine. Skywalker's sacrifice brings balance and redemption.
N'Kata Del Gormo was a Force-sensitive male Hysalrian Jedi Master who lived during the times of the Galactic Republic. According to legend, he found and trained Yoda and a Force-sensitive Human friend.
But with the sudden reveal that she was Palpatine's granddaughter, fans were left to wonder whether that was the original plan, or a last-minute decision. And answers were finally given by Rey's actor, Daisy Ridley, who admitted that Obi-Wan was originally Rey's father.
If Rey struck him down, Palpatine's spirit would pass into her, allowing the Sith to be reborn. Rey resisted him with the help of Ben, who'd returned to the light. But their defiance gave Sidious a new opportunity – he would drain the energy of their Force dyad to restore his own body, then rule alone.
It was the kyber crystal being attuned to the force that calls on Rey. So it so happen that the kyber crystal that calls to Rey was the one inside the Skywalker Lightsaber.
Ben grabs fertility beads off of her neck through their bond. When Ben brings her back to life he places his hand over her womb.
So let's look a bit closer at the legacy you're attaching your kid to when you name them Kylo: Our boy Kylo was born as one Ben Solo, the child of Han Solo and Leia Organa, named after the famous Jedi Obi-Wan (aka: Ben) Kenobi. Ben was trained in the ways of the Jedi by his uncle, Luke Skywalker.
According to ComicBook.com, the novel reveals that Ben had some final words for Rey in one of the film's last scenes. “A voice came to her through the Force, clear and strong. 'I will always be with you,' Ben said. She smiled.