Isn't this a question of verisimilitude? It's one thing to consider your character non-binary, but it's a flip being switched for the whole world, not only that, your character might be the only non-binary person in the whole game, yet for everyone it will go by without comment. At times is might strain credulity.
I know so little about Dragonborn that for all I know, they could reproduce asexually. Do they lay eggs? Does each sex have secondary sexual characteristics, it wouldn't be a matter of people calling them 'they' it would be a matter of a race of people who might not even have gendered pronouns. All Dragonborn might be 'He', because that's what they're consider the best translation from whatever language they speak....Draconic? This is reminding me of a subplot in Discworld.
