Although Star Wars: The Rise of Skywalker concludes with Rey choosing the name of Skywalker and completing her path to becoming a Jedi, it's possible that she was actually the reincarnation of Anakin all along. This question is the essence of one of the most compelling fan theories about the Star Wars sequel trilogy, that the name Rey chose for herself was already hers to use. Viewed in this light, a lot of things in the Star Wars sequel trilogy take on a new meaning.
Introduced during Star Wars: The Force Awakens, Rey quickly shows herself to be capable in hand-to-hand combat, skilled in piloting, and naturally powerful in the Force. In Maz Kanata's castle on Takodana, a lightsaber calls to Rey, giving her a Force vision and marking the beginning of her journey to defeat Palpatine once and for all. Her story leads her across the galaxy, and sees her torn between her future and past while she grapples with the pull of the dark side.
The idea that Rey could be Anakin Skywalker reincarnated may seem outlandish, but some viewers discussed it even when The Force Awakens was newly released (via Reddit). In Star Wars Legends, reincarnation is among the skills which powerful dark side users could learn, although this is not currently canon. However, if Anakin learned this skill as a Sith and was later reborn as Rey, it would explain their many similarities. The two most notable things from Star Wars: The Phantom Menace are that, with no formal training, Anakin is a supernaturally skilled pilot and he can wield the Force intuitively, using it to repair his pod racer while in flight. During The Force Awakens, Rey shows the same two talents, piloting the Millennium Falcon to evade First Order TIE fighters, and using a mind trick to escape capture.
The lightsaber which calls out to Rey is Anakin's. In Star Wars canon, the kyber crystals which power lightsabers form Force bonds with their users, meaning that saber was already bonded to Anakin and, by extension, Rey. This is why Rey could find her focus while wielding it against Kylo Ren. It's also why the lightsaber's Force vision ends in the voice of Obi-Wan Kenobi, Anakin's former master, saying, “these are your first steps.”
Themes of patience and balance are scattered throughout the sequel trilogy, particularly during Star Wars: The Last Jedi. The temple on Ahch-to shows a mosaic floor depicting a Jedi in meditation, between light and dark, reinforcing the idea that Anakin is still waiting to fulfill his prophecy to bring balance to the Force. With the revelation that Emperor Palpatine survived the events of Star Wars: Return of the Jedi, this task was still incomplete. Palpatine's eventual defeat by Rey's hand gains new significance as the completion of Anakin's prophecy. Similarly, even as Kylo Ren vows to finish what Darth Vader started, Anakin's absence is notable during the sequel trilogy. Return of the Jedi saw him turn away from that goal, choosing his family over Palpatine's quest for power. At the end of the original trilogy, Anakin's appearance as a Force ghost confirms his return to the light. So why would he not appear to Kylo Ren, his grandson, to tell him this? A simple answer is that he cannot, because his spirit is alive, as Rey.
This theory may never be confirmed, but it certainly gives new meaning to the title, Rise of Skywalker. It's a shame that there wasn't a scene between Rey and the ghost of Luke, mirroring the scene between Luke and the ghost of Obi-Wan during Return of the Jedi. Hearing Luke telling Rey, “no, you are my father,” could have easily become an iconic Star Wars moment. Perhaps instead of Rey's new lightsaber being yellow, a better choice would've been a balance between Anakin's blue and Darth Vader's red. Perhaps Rey's new lightsaber should have been purple.