Look, this isn't about real life transgender people. I feel like a few in this thread are getting worked up and feeling attacked, that's not the case. It's very much logical for children to be raised communally by trained professionals vs parents. A parent raising and shaping and loving their own child, that's an emotional idea (and one that I agree with) but it doesn't guarantee that each child will be raised properly. Some parents will be abusive or neglectful, some will be too lenient or too strict simply be sub par and they will each have a different level of education. Raising them communally would let you start the brainwashing young and make sure each child is as close to the perfect worker bee as you can get. In a thread on this same subject that was made soon after the game came out, David Gaider's replies on the subject didn't lead me to believe that Aqun Athlok were actually transgender, but were instead random biological females who were good at fighting and relabeled as men(whether they liked it or not) as a loophole so they could become soldiers. But also, why would a society that won't let you keep your children, won't let you express romantic love, won't even let you choose your job let you live as something other than your biological sex? Would someone who grew up in a society where the only difference between genders was physical and reproductive even identify as another gender? If there are no gendered clothings, attitudes, mannerisms, social expectations, emotional states, etc...then why? You're looking at it through the lens of our real society rather than this made up one.
"Look, this isn't about real life transgender people. I feel like a few in this thread are getting worked up and feeling attacked, that's not the case."
You aren't saying anything offensive about transfolk so I doubt anyone would. It's been pretty civil given the subject matter.
"A parent raising and shaping and loving their own child, that's an emotional idea (and one that I agree with) but it doesn't guarantee that each child will be raised properly."
Neither does raising them communally. I'm honestly trying to see how not having your actual parents take care of you is objectively better; that's something too subjective to determine.
"But also, why would a society that won't let you keep your children, won't let you express romantic love, won't even let you choose your job let you live as something other than your biological sex?"
Romance isn't something they forbid, so much as less interested in due to their obligations to the Qun. But to answer your question, it solves the inevitable dilemma that comes with a child being evaluated and excelling in a role normally associated with the opposite gender. If - and this is speculation of course - an Aqun-Athlok is in fact trans in the way we'd think of it, then we should remind ourselves that they probably view their masculinity and femininity within the context of how their society constructed it. That includes behaving, acting, and pursuing things associated with the respective gender. It's not a choice, but an obligation, as each person has an inherent purpose in their beliefs. Let's remind ourselves that Aqun-Athok are literally viewed as men or women and would likely be treated exactly like a man or woman would, as well as expected to perform the roles and duties of said-gender; it's the only way their teachings and beliefs about gender roles can remain true.
"Would someone who grew up in a society where the only difference between genders was physical and reproductive even identify as another gender? If there are no gendered clothings, attitudes, mannerisms, social expectations, emotional states, etc...then why? You're looking at it through the lens of our real society rather than this made up one."
This is all speculation, and you contradicted lore. In the Qun, there are very clear attitudes and social expectations men and women have; the very roles they've been given boil down to essentialist beliefs about the duties of that gender. We don't know much else about the Qunari to talk without conjecture.
Also, virtually every society in Dragon Age lore is based on or reflects sociopolitical systems in real life, even down to the titles some of them have and the annoyingly anachronistic French bard. All we have are the lenses of real societies to draw examples from in order to gauge a basic understanding about these cultures. They're obviously not exactly the same, but in some cases are very similar, and I'd conjecture that unless it's noted explicitly, there are ways we can relate to them based on what we've experienced ourselves.