omfg Avatar shipping is such a minefield. The main overlap I see between the Zuko/Katara people and the Harry/Hermione people is the focus on coded symbolism and the strong identification with the main female character. (Which I also see in Ron/Hermione people, although not so much with Aang/Katara people for whatever reason.)

I have to admit that I never saw the appeal of Zuko/Katara outside of pretty pretty fanart, but that's because I fell in love with Mai's character and the way she interacted with Zuko. Aang/Katara is kind of a weird ship for me because while I think it's adorable and love the way it fits into the overarching plot, on a purely character-driven level I've really come to love Aang/Toph more, especially now that the series is over and I can look back and see how everything played out.

I suspect I have a weird perspective on this because while I like Katara a lot, she's probably my least favorite of the main characters (that'd be Sokka, with Zuko and Toph close seconds), and I suspect that's kind of a minority opinion among shippers and the fandom in general.