Children have a tendency to bond closely first with their opposite sex-parent, and only afterwards with their same-sex parent. Disruptions in this course of childhood development tend to have pervasive effects on the personality of the person in question.
Aang As a member of the Air Nomads, it would seem that Aang was raised by the male (as evidenced by several flashbacks throughout the series) and female (as evidenced via flashbacks in "Appa's Lost Days") monkhood as a whole, never knowing who his biological mother & father were and thus coming to treat the religious orders of his people as his mother and father figures (for the second, one monk by the name of Gyatso was prominent in his life). Since it would seem that it was due to his failure to follow up the paternalistic demands the monks (with whom he seems to have spend much more time than the "nuns") placed upon him that the Air Nomads were killed off, he would therefore feel over-burdened with trying to prove himself as a full member of Air Nomad culture by living up to their ideals in a world where they no longer existed. This could easily have devolved into a personal, merciless bitterness towards the unkind world which had abruptly cut him off from the gentle, spiritual setting of his childhood (as we see inklings of throughout the series), in his attempts to live up to their optimistic, high-minded ideals despite such problems, had not he been wrapped in the caring, maternal arms of Katara soon afterwards (in a literal, and also a more metaphorical, sense).
Katara Katara had Kya removed from her life early on; she seems to have responded to the disruption this caused for her existing father-daughter relationship with Hakoda by setting up her mother (and thus her mother's relationship with her father) as a heavenly ideal goal which she ought to try her hardest to achieve in her own life. Too soon afterward, her father in turn left her to her own devices until the War's end, to which she seems to have reacted by eagerly hoping for the bitterness of the War to come to a close so that her idealized goals could then be brought to fruition and became willing to endure any hardship to overcome such problems. Her strong desire to learn Waterbending would be a subset of this psychological orientation; since it was her lack of Waterbending ability which cause her father to reach out for a teacher, with all the sorrow this eventually brought them through the person of Yon Rha, in her mind it would be through compensating for this via struggling to learn the art that the heavenly ideal of the life they once lead might be brought back down to earth, As can be observed from "The Southern Raiders", she could quite easily have slipped into an arrogant obsession with her underlying forebodings that her optimistic compassion towards the rest of the world had died in the form of Kya for a reason - due to her own "faulty" difference from her mother as a person who could not control her gift in order to protect others - had not Aang (with his oft heroic-seeming attachment to his responsibilites as an Air Nomad) been there to indicate that her mindset was something needed for a larger (spiritual) outlook on the universe world and was in now way in opposition to the way the world ultimately worked because of some supposed faultiness as a person on her part.
Why The Two Bonded As Closely As They Did In "The Awakening", the way Katara had set her hopes on a return to the idealized understanding of their home life before the War had been brought to their shores is made plain for the audience to see, with the Waterbender associating the persons of Hakoda (her father) and Aang (her blossoming love interest) in her mind and feeling wretched at the fact that both had broken her pyschological attempts to try to return herself to her home life beforehand. Katara is drawn inwardly towards Aang in the intimate understanding that he considers it his responsibility to preserve the memory of the high-minded ideals of the Air Nomads that once actually existed in the world, which gives her hope that life may yet reflect the one she once wished to return to in a growing, dynamic manner which wouldn't be abruptly taken away as her life before Yon Rha had been. Aang, in turn, is drawn inwardly to Katara in the intimate understanding that she is motivated to live up to a personal ideal of compassion often at odds with cynical operation of the larger world during the War, which provides him with a sort of maternal & thriving sense of optimism and compassion l he would otherwise forever be missing since the passing of the Air Nomads and reaffirms that paternalistic demands of the monks may yet be lived up to. Aang helps Katara to not doubt herself because he himself has had to overcome similar doubts himself, and Katara preserves Aang from becoming bitter by soothing the strain the genocide of the Air Nomad people has upon him, empathetically dealing with his issues and insecurities with the optimistic sense of compassion by which she overcame much of her own.
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