The competitive aspect? Males in plenty of species compete for status and low status males are getting chased away from the pack, so evidently genes are expressive enough to encode gender norms.
You probably believe that your views about genders is coming from observations of "males in plenty of species", but have you considered that maybe your views about "plenty of species" could come from your views on genders?
How is competition between males a consequence of sexual dimorphism?
Note that males in species with low sexual dimorphism compete for mates as fiercely as males in species with high sexual dimorphism.
Also note that the human species has very low sexual dimorphism relative to other species. E.g. our females are not many times the size of males, as they are in Black Widow spiders, our males don't have different colours of body coverage, like many species of birds do, etc etc. And of course many humans can easily pass for the other sex by changing their hair-style and clothing as evidenced by numerous cases in history where females passed as males by cutting their hair short and wearing pantaloons etc.
