I'm going to ignore this debate on racism within film production and going to stick with the fundamental question that Lohrenswald presented (since he encouraged me to post this after discussing this thread with him on IOTchat). It's an interesting conversation, don't get me wrong, but it's kind of off topic. Maybe you should consider a new thread?
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I think the biggest issue people Lohrenswald and co. have when asking this question, they take the meaning of "identifying" too literally. In general, art is a lot more visceral and abstract than even other humanities like history; a lot of it is just based on raw human emotions and concepts that people feel but can't nessecairly put into words. It can be hard to identify what we actually like and dislike about aspects of art; because it's not an exact science and there's always a lot of nuanced variables, along with the uncertainty of human emotion, that prevent universal theories of story or characterization conventions.
Hence, in art discussions, i feel like people tend to use terms, not incorrectly, but different to how we'd normally use them, in order to cover for these experiences that we most definitely feel but can't actually explain. "Identifying" is one of those terms; as people in this thread imply, people think the term is a literal synonym for empathizing. And while empathy plays a large role, I'd argue that being able to identify with a character does not mean you empathize with him or her, or vice versa. Instead, I believe identifying in the context means more "does the character have internally consistent humanlike qualities that I can recognize and understand, if not agree to?'
Now, I want everyone here to think long and hard on who their favorite character is. In the case of myself, it would definitely be Lucina, who I love so much that I use her as my avatar here next to all my posts. Now, what about Lucina endears me so much into her? I mean, appearance is always a factor when it comes to characters. In the case of Lucina, I'd even argue that her appearance heavily plays into her characterization, especially when it's so heavily tied to her identity in the story (makes more sense if you played Awakening yourself).
That being said, I could easily have conceptualized Lucina, with her exact appearance and name, and even the same general concept of a character (time traveling swordswoman who constantly seeks affection and approval from her family/father), but if I made how she interacts with the world completely different, then this new Lucina would be in herself a completely different character as well. Perhaps the changes would be better, perhaps the changes would be worse. I couldn't say for certain, but they'd be different.
To me, Lucina has three defining traits, besides the high concept that I described in the last paragraph that endears me to herh. The first of this, is her relationship with her friends and family. It's obvious to any player that she highly values her family, especially her father, Chrom. Even when the audience is unaware who she is or that Chrom (who is the main character of the story) is her father, she praises that Chrom taught her everything she knew about fighting even as she proceeds to duel him.The second is the latent trans symbolism that she's loaded with. That alone could really be an essay in itself, but as a tl;dr Nintendo is not a stranger to trans themes in previous works (see Mario 2 and The Thousand Year Door; hell there's a literal trans character in awakening itself) and while I don't make the claim Lucina is trans, she definitely is evocative of the trans life and struggle. The final is her demeanor, or the way that she acts and responds to the various issues that arise both in the background and within the actual story.
Now, can I directly empathize with Lucina? In some ways I can actually. I know how it feels to want to seek approval of your parents; I think it's a raw instinctual emotion that every human has to be quite honest. I can empathize with how she tried to hide her gender for the first arc of the game; I do that all the time in real life, and for a while I even had to do it here on CFC. I can empathize with trying really hard to be a stout, stoic hero that can take on anything, but having to stop sometimes to cry and let it out, because goddamnit if that doesn't describe my entire experience trying to transition...
However, there's just a lot of things I can't really empathize with her, but I can identify with. For example, no, no matter how hard I try, I can't empathize having my entire family being murdered by monsters, and seeing civilization itself collapse by their relentless assaults. But I can identify with that scenario on a more abstract level; seeing my entire familial relationship implode after they rejected who I am, being forced to live off my own in completly uncharted territory for myself. Yet, in both of our scenarios, Lucina and Inagree we have to fight on, because the alternative is even worse (Her case? The extinction of humanity. In my case? Likely suicide in a few years). Quitting is far easier, but is out of our options.
Likewise, I can identify with a lot of Lucina's demeanor, because utimately I idealize her philosophy, even if I never come to close to it in my life. Lucina, when given an immovable object, becomes the unstoppable force to break it down. Lucina doesn't believe in concepts like fate; her most famous quote is quite literally a contradiction of determinism, "I challenge my fate!" Instead, she believes any problem can be overcome, if you are resourceful and have faith in yourself. It may not always be easy (or practical in real word scenerios; there's no time travel irl), but Lucina will never give up and accept defeat. It's a mindset that I ultimately agree with, if not always rigorously stand to like she does.
And finally, while I notice and identify with all the trans symbolism that Lucina oh so clearly expresses, it's quite evident that Lucina herself is not trans. She expresses or even hints no gender dysphoria at any point of the story; it's all more on a conceptual level of how Lucina is designed and her role/arc within both the narrative of Awakening and her place in the series as a whole (it's a pretty complex theory; trust me you're better off just taking my word for it). Therefore, as much as I want to, I can not directly empathize with her on this specific issue.
So, to wrap this all up, being able to identify with Lucina so much made her into my favorite fictional character and made Awakening one of my favorite games. I think it would be safe to say I would not have enjoyed the game anywhere near as much if Lucina did not appear in the game. Does that have to say that I have to identify with a character to like its story? No. 1984 is one of my favorite novels, but the characters are actually not that important in the grand scheme of things. 1984 was always more about the political message that was interwoven between the setting and characters, which were only an entertaining vessel to express it. But most works of fiction are not 1984; they are not written for the expressed notion of a political tract. In that case, having strong, identifiable characters is what really gives your fame, or movie, or book, that long lasting appeal.
TL;DR Identifying means strong characterization, strong characterization means strong story