It’s not her identity that’s the issue, it’s her own acceptance of that identity. Earlier in the film, there’s a scene where an alien child asks her what her family name is and she says her name is “just Rey” in a sort of sheepish, slightly forlorn tone. At the end of the last film, she came to the realization that she’s a “nobody from nowhere” but we can see in this scene that she’s still not quite comfortable with that newfound knowledge. It’s still a point of embrassment rather than pride. By taking it as a badge of honor at the end of the movie, she’s fully actualized and completed her arc, whereas taking the Skywalker name feels to me like it’s a step backward. It’s her clinging to an idealized version of the Skywalker legend as she did in the first half of the Last Jedi. She needs to be comfortable being “Just Rey” before she can be “Rey Skywalker”.
These are all just my opinions of course, but I cared a lot about this character and was disappointed with the direction they took in tieing her identity to other people, whether it’s the Skywalkers or the Palpatines. As we see in the prequels, the Skywalkers and the Jedi as a whole have a complicated, messy legacy and while Rey learned from and was inspired by them, she could have also been the start of something new.
Perfect analysis. Couldn’t have said it better myself.