I don’t know but have been wondering about this too. It seems he took Liliths name but I’m not sure if it means something else

I think whether or not it’s his or Lilith’s last name, he used it when he was on the isles for a reason. Guessing because he was studying feather tails and that wasn’t a highly suggested research topic. I think if he’d used the last name Sorrengail, it would somehow get back to the continent that he was researching/writing something he probably wasn’t supposed to.