Anon asked: That leaves the obvious follow-up question: what is it about Azula that makes Zuko want to claim her?
Literally everything about her.
He was jealous of Azula's talent for… well, everything, but the fact that he was jealous proves he did see her strength (physical and mental), her intellect, her ability to improvise, etc. And he 100% likes her sarcasm, mean comments, and her usually "cold" demeanor (there's a reason that, of Azula's two friends, he was attracted to the one most similar to her). The only reason he ever "hated" her was because he was forced to compete against her. Take away the rivalry, and you'll realize that there's literally no other person who admires Azula as much as Zuko does.
There's also the fact that Azula understands him and wants his company (not always with the greatest intentions, but still). She always knew just what to say to get whatever reaction she wanted from him and what buttons to push and what strings to pull to make him act according to her plans. While that could often end up screwing him over, it is also radically different from every relationship he had - Ursa understood and loved him, but disappeared suddenly; Iroh loved him and never left his side, but he wasn't always able to get through to him because he didn't fully understand Zuko's wants and needs; Mai loves him, but they have trouble connecting and she is prone to shutting people out; Ozai knows how to manipulate him but he has no idea who Zuko is as a person and has no interest in ever finding out (there's a reason the emotional, tragic music was used when Zuko was confronting his sister, not his father); his relationship with the Gaang is so messy it took three books to sort it out.
Azula on the other hand, wants him around, both because she sees him as her plaything, and out of genuine love (though she won't admit that part), even when it could actually go against her goals - she wants to be Fire Lord, yet she keeps her promise and bring him home instead of having the Dai Li imprison him after they take over Ba Sing Se. And the reason she was the one who made him give up on redemption and come back to the Fira Nation was because she knew him well enough to see that he was still a proud prince who wanted back all that was his by right - something even Iroh missed because he chose to see solely the good in him, while Azula saw and understood all of his weaknesses and strengths.