I didn't get that he was her father(!) or her half-brother. He is her full-blood brother.
Their mother and their father were prob not married, because the dad is emotionally dead inside (as we see). He had "Miller" with her, hung around a bit, Miller knew who his dad was, they didn't live together as a family, dad supported them financially.
Mother had Lilly. She did NOT die in childbirth. That was a lie dad told Lilly. For whatever reason (and we will prob never know now that they stupidly cancelled the show), mother kills herself (she was NOT murdered - Miller just sees it this way because he is an mental/emotional basket case. He wants someone to blame because it is a traumatic experience.), dad sets up financial support for the rest of Miller's life (out of guilt for abandoning him, out of obligation for being his father, we don't really know).
Miller is not the most mentally together guy. He has never had real relationships, he is a liar (all his false IDs, etc.), gambling addict, pervert, etc. He does not know how to relate to his sister. Does he love her, and/or is he jealous of her life with dad and hates her? He steals her panties (which say "Night, Night Kitty" not "cat" and wears them because he is a confused guy. It doesn't necessarily mean he really wants to be all incesty with her. He's been alone all his life. He is upset, and a little crazy. He wants her to love him. He wants a sister, but doesn't know what that feels like. So, he goes into his male, pervert, default self. He wears them to see what it feels like to BE her, to be CLOSE to her.
And as far as the panties themselves... Lilly says that she basically only loved her cat. She is a young, sexy girl, that longs for intimacy and has no outlet for it. So, the panties are a naughty play on that - the slang for a woman's private area and her little-girl-lost love for her cat. She's pretty dark and quirky, so the humor of the panties makes perfect sense to me.
They say Night Night Cat not Kitty.
u forget Lily seems to also live a lonely life mostly because she isolates herself and really seems to hate herself and feels nobody will miss her when shes gone