Zuko is canonically a sappy hopeless romantic who enjoys romance stories like Love Amongst The Dragons, we know this because his problem wasn't that his mom would take them to see a gushy romance play but that they did it badly. You don't care about something getting butchered unless you like that thing.
He has trouble expressing himself but tries to make thoughtful gestures and give little gifts to people he cares about to show affection.
Early on when Iroh and he are living in a cave and he's going through one of his Robbing People With Swords phases he steals a fancy tea set for Iroh in a misguided attempt at showing affection.
In another misguided attempt at affection he gives the Earth Kingdom boy his knife.
After struggling the entire night to bond with Jin because he's socially inept he does his cute little gesture of lighting the lanterns and risking outing himself as a firebender because he wants to make her happy. And then gives her a coupon for a free cup of tea when she clearly wants to kiss him, because he's clueless.
We see this style of showing affection come out with Mai as well except she sneers at it and shuts him down and hurts his feelings about it. He is the kind of sappy bastard whose first instinct on seeing something pretty is "I'm going to give this to my girlfriend :)" but it seems like this is supposed to be showing us that he and Mai aren't a good fit. He tries being sweet and vulnerable and makes little gestures, like seeing a beautiful perfectly intact conch shell and trying to give it to her, and she verbally rolls her eyes at him and hurts his feelings about it.
Later on when he's with the gaang and starting to feel safe being his true self, he's showing them affection in his typical way by making and serving everyone tea (and on a more subtextual level I think this is a way to demonstrate humility to them, since he comes from royalty he's trying to show that he sees them as equals).
He does other kind gestures to show his new affection for them in the forms of leading a prison break with Sokka and taking Katara on a trauma closure (but murder not NOT on the table if she wants) mission and waiting outside her tent all night like a wet dog on her doorstep.
Dude is a piner and a yearner. And you know what, you know who would LOVE IT if Zuko found that seashell and gave it to her and wouldn't turn up her nose at it? A certain waterbender who he acts more in love with than his canon gf.














