If nothing else, it’s because Kylo never reveals her true last name to her (if she ever had one). The only thing she has going for her is the fact that Palpatine claimed her as his own at one point, so she is free to go by Rey Palpatine if she chooses.

In my mind, though, the idea is that the Skywalkers also have the right to be claimed by Palpatine as his own, if that is what they want, and use his name. But, ultimately, that would be a corruption of the truth, a Palpatine specialty. I’m hoping that it’s clear in an edit like this that Palpatine is only claiming what the Force made for himself, and the reality is that Rey gets to decide her own legacy. In reality, the Skywalkers always belonged to the Force itself, not him, so Rey takes on their legacy.