Axa

There is a saying - "When in Rome, do as Romans do"
This, I speculate, also applies to social gender roles which are differently realized in various cultures. So, yes, your dressed-up monkey metaphor is appropriate, but it is not directly connected with your prior statement that there is no true gender. Please define what you meant by the term "true gender".
Currently it seems to me that all genders are true within their respective environments - within the group that has invented and accepted them as being true.