Can I choose my own gender? Are genders even in this game?

 
You can only choose your partner's gender, by selecting Cool (male) or Cute (female) during the personality quiz. Your character defaults to male. But don't worry too much; this doesn't affect any dialogue, and only really matters if you have visible gender differences (apologies if you wanted to be a female Pikachu, I know how you feel).
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Genders don't seem to play a significant role. They are not obvious during gameplay like they were in the Explorers games, except for visible gender differences. (Attract works on everyone, and Rivarly appears to work differently as well – currently unsure about the details.)