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.
Zuko Zuko's early home life was disrupted when Ozai (in a web of political intrigue) took the throne of the Fire Lord and tore asunder his son's relationship with Ursa by banishing her from the nation. In the absence of his mother, Zuko obviously attempted to compensate for his apparently politically inappropriate relationship with her by seeking to hold himself to the cold-hearted, cynically-minded "politics only" demands of his father; as recounted in "The Storm", this proceeded to get him banished, but which he continued to try to act upon even in banishment. Eventually, Zuko reached the point where the starkness of his father's political cynicism was revealed (as recounted in "Nightmares and Daydreams", as well as "Sozin's Comet, Part 1: The Phoenix King"), contrasting too sharply with the more compassionate setting in which he had originally been raised for him to not feel alien from himself had he attempted to hold himself to such standards against what his experiences in banishment had informed him about the rest of the world. At the very end, however, Zuko was not necessarily completely stable in his turn away from his father's cynical worldview; having never been assured that Ursa's banishment to save his life was not made necessary by an innate opposition between her more compassionate nature and the political requirements placed upon royalty, his being made Fire Lord after his father could easily draw out these lingering insecurities until his mother is returned to him and/or if in some other way her maternal nature is revealed to him as ultimately serving to build up the royal family in the following of its proper duties (rather than existing in a permanent opposition with it).
Mai At an indeterminate time in her childhood, Mai's father began seeking to climb the Fire Nation political ladder. Her mother reinfoced the tendencies of a political focus to weaken strong familial bonds by expecting Mai to remain passive so as to provide as little disturbance to her father's goals as possible, rather than trying to sway her husband to try to co-ordinate his political life with his family life so that the former wouldn't have to impose so strongly on the latter. Mai seems to have reacted to her father's placing of their father-daughter relationship under the thumb of Fire Nation politics by withdrawing from the rest of the world emotionally (including, of course, from her family, as shown by the causual-ness in which she accepted Azula's decision to not exchange Bumi for her brother in "Return to Omashu"), her concentration on violent weaponry probably developing as a reflection of how the Fire Nation's militiant politics imposed itself on her pyschological development and becoming her main psychological outlet by which she kicked back at the world at large.
Why The Two Bonded As Closely As They Did From her demure way of acting in the flashbacks present in "Zuko Alone", it is evident that Fire Nation politics had already intruded into Mai's life as they had not yet done in Zuko's, though there were already bad omens present in the way the Fire Nation royal family was beginning to operate at that point. Because of this, Mai is drawn towards Zuko inwardly in the intimate understanding that he is in danger of becoming as her father did but had not yet reached that point, pyschologically motivating her to bond with him as a way of overcoming the effects her poor relationship with her father had upon her life. Zuko, in turn, is inwardly drawn toward Mai in the intimate understanding that she is in danger of fading from the miminum of attachment to one's own emotions required for pyschological stability because of the imposition of Fire Nation politics upon her life but is yet able to show concern for him and the way his family's falling apart affected him nonetheless (at time, to a greater degree than one might reasonably expect from someone in her situation), offering hope that one like his mother could operate without detriment to the necessity of politics for the Fire Nation royal family.
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