The Emperor insisted Rey was his granddaughter in Star Wars: The Rise of Skywalker – but Rey Palpatine was never really a Palpatine at all. The mystery of Rey's parentage ran through the entire sequel trilogy, from pretty much the first moment Rey was introduced in Star Wars: The Force Awakens. She finally learned the shocking truth in Star Wars: The Rise of Skywalker - even if she did ultimately choose to reject both her genetic family and her Palpatine name, claiming the mantle of a Skywalker instead.
According to the Emperor, Rey is, in fact, a Palpatine. At first, readers believed this meant Palpatine had a secret family – a lover, children, and, subsequently, a granddaughter. But the novelization of Star Wars: The Rise of Skywalker revealed Rey's father was actually a clone, created as a host for the Emperor but found to be lacking in the Force. More details came in the 2022 tie-in novel, Adam Christopher's Shadow of the Sith, featuring Anakin and Luke. However, Rey's history as covered in the movies already confirmed the adoptive Skywalker was never a Palpatine at all.
Clones and children aren't the same thing (despite Jango and Boba Fett's relationship). The best parallel is with Cut Lawquane, a clone trooper who abandoned the Grand Army of the Republic at the height of the Clone Wars. Cut Lawquane settled on Saleucami, marrying Suu Lawquane and fathering two children, Shaeeah and Jek; Clone Force 99 paid him a visit in Star Wars: The Bad Batch, and although the Bad Batch considered Cut Lawquane part of their family, there was no suggestion his children were considered Jango Fett's grandchildren. By the same logic, Rey being a Palpatine isn't quite accurate. Here's why "Rey Palpatine" isn't Rey Skywalker's name and never was, as well as why her claim to Luke and Anakin's name at the end of The Rise of Skywalker was right.
Rey's Father Was Palpatine's Clone, Not His Son
It's certainly in character for Emperor Palpatine to claim possession of something or someone to which he has no right. He discarded his cloned "son" Dathan when he learned his progeny lacked Force potential and could play host to his dark side essence. He only became interested in claiming this supposed "family line" when he learned of Rey's existence. Lucasfilm's Secrets of the Sith expands upon Palpatine's views regarding his clones, in notes from Emperor Palpatine himself. "I rejected this useless creature, but I chose to let it live," he reflected. "Even if this 'son' was undeserving of my legacy, I surmised that the Palpatine blood rushing through its veins might someday become useful." It's interesting to note that, in his private records, even Palpatine placed quotation marks around the word "son." He knew full well he had no real claim to consider the clone as part of his family.
Rey's father was a Palpatine clone named Dathan, not Palpatine's son. Rey was never really a Palpatine, then, even by the private admission of Palpatine himself. Regardless, in every aspect save genetics, she has truly become a Skywalker. Star Wars has redefined what it means to be a Skywalker, revealing that Skywalkers are agents of balance – beings chosen by the Force to help retain a healthy equilibrium between its Dark and Light sides. Anakin accomplished this when he struck a powerful blow against the Sith in Return of the Jedi, and Rey embraced his legacy when she, too, opposed the Emperor. She rightly rejected being a Palpatine and justified the name of the final film in the sequel trilogy, Star Wars: The Rise of Skywalker. If nothing else, her childhood scavenging in the deserts of Jakku is far closer to Luke and Anakin's Tatooine upbringing than the aristocratic childhoods of Naboo's noble House Palpatine.
Palpatine Saw Rey As A Tool, Not A Granddaughter
Kylo Ren idolized his grandfather Darth Vader, but Rey never saw Palpatine in this way, never even acknowledging her familial connection to The Emperor as anything beyond a genetic technicality. For his part, Palpatine didn't really think of Rey as family either. In fact, the idea of a clone's daughter taking the name Rey Palpatine would probably offend him, judging by his tone when speaking on Dathan. There were no parallels with Darth Vader wanting to rule the galaxy alongside Luke. Palpatine wanted to take over Rey's body and use her to continue furthering his Sith goals, just as he'd intended with her failed clone father. Palpatine used their connection as emotional leverage when trying to tempt Rey to the Dark Side, but that's pretty much the extent of her role as a granddaughter. It's unlikely that the sociopathic Palpatine can even feel familial love. He's so entwined in the Dark Side that, in an incredibly dark moment for the Disney Star Wars canon timeline, he killed his own father and slaughtered most of the Palpatine family. Why? To prove to Darth Plagueis he'd make a worthy apprentice. The idea that such a man could consider anyone as a family member, much less the child of their failed clone, is highly doubtful.