tl;dr is at the bottom

Drawing on some research on the subject, it looks like people naturally observe and imitate each other and it is isn't an entirely conscious/logical process. This is obvious in the Asch line experiments and Zimbardo prison experiment. People made statements and took actions that they knew to be wrong because they were following group influence.

However, the process runs quite a bit deeper than logical consideration and doesn't just affect adults. As the first link shows, children (even pre-schooling age kids) will actively develop and enforce norms, even illogical ones. Another study shows they do this with games and another one demonstrates they will, unbidden, complain when anyone breaks a norm.

Very simply put, I think the science shows that we cannot live free of social norms. As a result, the idea that "you can be whoever you want" is almost certainly false and will not stand the test of time.

The next question is do men and women have to have different norms? Looking at Manhood in the Making (David Gilmore) which is a collection and summary of ethnographies on masculinity, in rare cultures it is possible for men and women to be seen as substantially similar, but even here there is definite differentiation.

It is worth noting that for men and women to have similar personalities, the examples in Manhood in the Making all force men to be soft, effeminate and relatively passive (IMO). My impression from other readings is this is not the case in egalitarian societies. In general the closer a society gets to egalitarian, the stronger men become.

tl;dr / conclusion - It appears pretty clear the promise of "you can be whoever you want" is a false promise--humans simply don't work that way. From a very young age we spontaneously create and enforce norms. The idea that we couldn't or wouldn't is simply wrong. We can probably force men to be soft and similar personalities between men and women, but unless you believe personality must be identical, I do not believe it is necessary for egalitarianism.