Like The Rise of Skywalker as a whole, the last scene of the sequel trilogy is pretty controversial. A lot of people don't like Rey taking the name of Skywalker, for various reasons- some feel that it doesn't fit her character, some think that it undercuts certain messages about family, and some people just think the line was corny. Regardless, I still like that ending for Rey's character, and I have some reasons why.
Maybe you already agree with me, maybe you don't, but I hope you'll at least give me the opportunity to explain my point of view.
To begin with, I want to address one of the most common (and valid) criticisms of Rey Skywalker- "she and Luke didn't even like each other, why wouldn't she have chosen Solo or Organa?" Which is definitely fair! But I don't agree.
First of all, I think people are a little dramatic suggesting she didn't like Luke. While there was definitely tension between the two in The Last Jedi, they still ultimately did share some good moments on Ahch-To, she was saddened by his death at the end of The Last Jedi and her scene with his Force ghost in The Rise of Skywalker showed that there isn't animosity between them. She still clearly regarded him as an important guiding figure for her journey.
But still- why not Organa? She did get along better with Leia, and spent much more time with her.
And to that, I would say that's because Rey just doesn't fit the Organa legacy. I see Rey taking a name less as a look back on her life and the people in it- she did plenty of that her whole life, obsessing over her childhood and parents- and more of a look forward towards what she wants to be. She's deciding which legacy she's going to live up to.
The Organa legacy isn't one of Jedi. Yes, Leia trained as a Jedi and taught Rey, but that's not her legacy. The Organa legacy- that of Bail, Breha and Leia- isn't one of Jedi, it's one of leaders. Fighters, people who inspire in wartime and people who manage and protect during peace. If Rey is claiming a legacy that she wants to move forward into, the Organa legacy just doesn't fit that. The Organa legacy is a good, important one, but it isn't who Rey is. Rey isn't a senator, general or queen like the Organas. She's a Jedi- like a Skywalker.
When Rey says that she's a Skywalker, to me that means that she's deciding her own future. She's deciding for herself that she wants to be, she wants to live up to the Jedi legacy that Anakin and Luke left behind (more Luke than Anakin because of the whole... child murder thing). Rey's life has been fraught with people making decisions for her, telling her who she is, and she's not going to live by those terms anymore. Rey is holding her ground, saying "I'm a Jedi, I'm a Skywalker, and that's who I am, that's who I want to be." She's looking ahead instead of looking behind- creating her own family and story rather than waiting for people from her past to come back for her.
I think it's a really good moment for many more reasons too. It shows a lot of growth for Rey and her relationships. She went from the lonely scavenger on Jakku counting down the days since her parents left, to building her own family among the Resistance, from calling herself a nobody from nowhere to proudly proclaiming herself with a name she's connected to those she loves and to the legacy that inspires her.
It's also, to me, a show of the important message that your "bloodline" means nothing when it comes to who you are and what your future is. That's something I know a lot of people liked about The Last Jedi claiming Rey as a "nobody." I do think that The Rise of Skywalker continues that message, albeit in a different way (one more akin to the Luke and Vader connection), and I think that Rey being "adopted" into the Skywalker family with the blessing of Luke and Leia is another show of that. Rey is no less a Skywalker than anyone else just because she isn't the biological daughter of any of them- they're all family because of the love and bond that they share.