Kylo Ren is not a Skywalker. We do not claim him. Next
His father was a Solo. His mother was an Organa. He never was, and never will be a Skywalker.
So Leia doesn’t count as a Skywalker, despite being Anakin’s biological daughter and Luke’s twin sister.
Good to know.
Nope, he’s not a Skywalker because he doesn’t deserve to be one for acting like a goddamn clown on the daily
And Leia is a Skywalker by blood but in her heart she is an Organa. You know. After the two adoptive parents who lovingly raised her, because even after he redeemed himself she hates Anakin and can’t forgive him and therefore doesn’t want to be affiliated with him
He’s also not a Skywalker regardless bc Leia deserves a less shitty son so he doesn’t count
Pay attention
Leia is an Organa point blank period. Her father was Bail Organa, her mother is Breha Organa, they made her who she is – a fighter against tyranny.
“Ben” taking the Skywalker name would be a fucking slap in the face to Leia. Highlighting the blood of Vader over her family that was exterminated BY Vader… that’s supposed to be respecting Leia? Leia embraces Luke as her brother, but her father is STILL Bail Organa.
This is made very clear in the Bloodline novel where Leia thinks, “When she thought of her father, she thought only of Bail Organa." She doesn’t think of Anakin Skywalker as her father. He was also her torturer who was complicit in the deaths of her family and planet, and killed millions more besides.
OP is also right that Kylo Ren has forfeited any claim to the family name with his actions. Any ONE of his crimes in the movie, from mass murder to torture to patricide, is enough to get him cast out of the family. It’s actually very common for heinous criminals to be cut off from their families, and Kylo Ren's are depraved beyond measure.