Just Write had his own video on this that I haven't seen linked here.So I was watching Jenny Nicholson and she made a good point. They play up her taking up the Skywalker name. But her only real connection to it was Luke. Leia never identified as a Skywalker and she has as much connection to Han Solo and Ben Solo as she does Luke anyway - in fact, probably much more.
He's kinder or atleast more understanding than most others, but he brings up a similar point at this timestamp. She knew Han for like less than a day and all he ever told her was to get off her ass because the universe is bigger than some desert hole. Luke she knew for a few days, but he similarly mostly told her the Jedi were shit and she should give up on her dreams until she convinced him otherwise. But Leia, she trained her for a year. Leia is the one who actually taught her techniques, maybe told her some more affirming philosophies, etc. It all happened offscreen, but of the main 3, Organa was the one who was there for her the most. Also, tho, Leia's naming convention is a bit weird. Organa is her adopted parents name, Skywalker is why she is a Jedi and Solo was the name she took after her husband (which I know they divorced and he's dead, but....I would imagine she'd maybe want to reclaim it in remembrance. Maybe Organa-Solo).