So, as far as the canon goes, yes, Rey is considered a Skywalker. It's just like how any adopted child would be considered a member of the family that they were brought into. Depending on which Star Wars fan you ask, the identity of Rey will vary. Some would consider her a Palpatine because of her blood.
But that itself ended up not being the final revelation, as Rey was firmly connected as Emperor Palpatine's granddaughter (yes, the Emperor Palpatine) in The Rise of Skywalker. The culmination of that movie saw her reject that family and instead carry on the mantle of the Skywalker name.
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.
Although she was not a Skywalker by blood, Rey adopted their family name to honor the memory of the Skywalker twins, both of whom served as Rey's mentors in the ways of the Force.
The Emperor claimed Rey was a Palpatine in Star Wars: The Rise of Skywalker - but she was never truly part of his family, and even he knew it. The Emperor insisted Rey was his granddaughter in Star Wars: The Rise of Skywalker – but Rey Palpatine was never really a Palpatine at all.
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.
And in The Rise of Skywalker we finally learn that Rey is actually the granddaughter of Palpatine. Her parents—one of which is the child of Palpatine—abandoned her on Jakku to keep her safe from her grandfather who was aware of her power.
In addition to rejecting Sidious, Dathan married a woman named Miramir, whom he fell in love with. Together, they conceived a child, their daughter Rey, who was born in 15 ABY. Although he renounced the Sith Eternal and his donor, the Palpatine bloodline would prosper through him, just as Sidious predicted.
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 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.
In The Rise of Skywalker's novelization, we learn that Rey's father was a failed clone of Emperor Palpatine. He was initially created to be a vessel to house Palpatine and his Sith power.
Despite being enemies, Rey and Kylo Ren share a connection called a 'Force dyad' and eventually become romantically involved with one another. 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.
Star Wars: The Rise of Skywalker revealed that Rey had been a descendant of Palpatine all along.
But Rey responded that she was all of the Jedi. Aided by the spirits of the Jedi who had come before her, she used both Luke and Leia's lightsabers to deflect Sidious's Force energies back at him, destroying the Sith Lord once and for all.
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.
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%.
HUGE STAR WARS REVEAL! Rey RETURNS and is PREGNANT with Force Ghost Kylo Ren's Baby!?! Disney's biggest reveal at SW celebration: Force Ghost Kylo Ren knocked up Rey.
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.
Initially unknown to Rey, she forms a dyad in the Force with Ben. The dyad is an unbreakable Force-bond that makes them one in the Force, despite being born as two physically separated individuals. Thus, despite not being related by blood, Ben is Rey's other half, making him her "soulmate" or "twin of the Force".
While on the planet, when someone asked her who she was, Rey responded by saying that she was "Rey Skywalker." Although she wasn't related to them, both Luke and Leia had been mentors to Rey throughout the sequel trilogy.
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.
Even though Finn is once again Force-sensitive, and tries to tell Rey about it throughout the film, the film ends without the story addressing his abilities. While fans were left wondering about Finn's Force status, subsequent Lego films have confirmed it, and Finn is already on the road to becoming a Jedi.
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.
Triclops Palpatine (son) Kendalina (daughter-in-law) Ken Palpatine (grandson)
Rey is the granddaughter of Emperor Palpatine. She is not biologically related to Anakin Skywalker.