while i have recently just started, i did want to make just a small suggestion of adding a nonbinary option for the game. while i’m not sure about how much pronouns make a play in this game’s dialog, i thought it’d be a nice gesture to have the option there for those that don’t identify as male or female. 
Neither your starter or current lead pokemon’s gender is checked in any story scripts, to my knowledge. The only case of gender being checked outside of game mechanics like the infatuation status that I’m aware of is the little easter egg when you try to PM yourself.
While I’m all for more representation, all selecting your gender does currently is set the gender of your starter. A non-binary option would require the game to keep track of the player’s gender, which it does not currently do to my knowledge.
It’s definitely a bit confusing that the game asks for a name and gender but then uses them for entirely separate purposes, though. What it’s really asking you is “what gender do you want your starter to be?” and “what do you want your player name to be?” While your starter’s name is set to the same thing as your player name by default, it can be renamed like any other pokemon.