No, gender/gendered brains isn't innate. I don't have a circuit in my head that tells me to like pink because its for girls. however, there could be some differences between males and females in visuo-spatial cognition, verbal and reading skills and of course hormonal differences that could influence behavior. Males overall are more prone to aggression (presumably due to higher testosterone) and you can see that bare out in crime statistics. it should be noted that there have been multiple studies analyzing this and at the end of the day there is no significant difference between male and female brains.

here are some articles and their findings. I'm not sure how great these studies are or if they have been debunked since.

https://stanmed.stanford.edu/2017spring/how-mens-and-womens-brains-are-different.html

https://www.cambridgeindependent.co.uk/whats-on/the-neuroscientist-shattering-the-myth-of-the-gendered-brain-9086242/