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How the "Surrogate Sibling" argument fits better for Aang/Zuko's relationship than it does for Zutara, a thread (warning: it's long!):
First off, yes, Katara and Azula are character foils. But this doesn't correlate to the way that Zuko perceives his relationship with them. Zuko has ALWAYS seen Azula as superior to himself. Someone that he could never catch up to in skill because she's a prodigy.
With Katara, it's not the same. When he first meets her, she accidentally freezes Sokka. She isn't a bending prodigy (at first), it takes practice. Zuko has seen and acknowledged her growth in skill throughout the show. He views her as an EQUAL. Someone who also had to struggle.
Contrast that to when Zuko says (to an unconscious Aang), "Not that you would understand. You’re like /my sister/. Everything always came easy to her. She’s a firebending prodigy, and everyone adores her." He sees Aang as a prodigy; someone whose abilities are better than his
like Azula! Due to his resentment of Azula, he projects that onto Aang. Aang is younger than him but more powerful and skilled, exactly like Azula. He correlates Azula/Aang so much that he incorrectly assumes that everything came easy to Aang (not knowing Aang's background.)
Zuko feels incredible anger towards Azula that he cannot express because she's the golden child (and he's the family scapegoat/disappointment.) In CoD, Azula has imprisoned him, taken Iroh away, and forced him to make a choice when he really only has one option (to side w/ her)
Because Zuko is unable to take out that anger on Azula (due to their power dynamic being skewed in her favor) he directs all that pent up rage into going after Aang. Again, largely because Aang has Azula's traits.. the one's Zuko lacks and were constantly used to demean him.
Even after joining the gaang, there's lingering hints of Zuko's resentment of Aang, particularly when he's struggling with firebending.
A: "That one felt kinda hot."
Z: "Don't patronize me!" (something azula probably did)
A: "Sorry Sifu Hotman."
Z: "And stop calling me that!"
A: "That one felt kinda hot."
Z: "Don't patronize me!" (something azula probably did)
A: "Sorry Sifu Hotman."
Z: "And stop calling me that!"
Zuko finally was given a chance to teach someone he considered superior to himself...to prove himself (much like in the Agni Kai when he mentioned lightning.) When he failed, it probably unintentionally reminded him of Azula. (Although sweet Aang was being genuine.)
But then in The Firebending Masters ep, Zuko chooses not to rely on anger and hate anymore and breaks this cycle of abuse. He shifts to encouraging Aang (someone he previously felt resentment towards) because he sees that Aang needs it. And Aang also gives him validation.
That ep changed their dynamic. Zuko is able to dissociate Aang from his feelings ab Azula. In a sense, Zuko is able to repair a pseudo-"sibling" relationship w/ someone who values and respects his skills, encourages him, and willingly accepts him. That person is Aang.
TL;DR: Aang is more like a sibling to Zuko than Katara based on the narrative. Zuko still "gains" a sibling w/o the intent of delegitimizing a ship (which let's be honest, is the goal of antis.) *Also Zuko/Azula should've been able to fix their bond so Aang is NOT a replacement!
