Gender has both psychological and physiological attributes and nobody is pure male or pure female in either category.
Psychologically, people can feel or act in a more male or more female way, irrespective of physiology. There is a certain amount of socialisation of gender and how each gender is expected to behave.
Physiologically, people have different levels of hormones at different times, and so perhaps the levels of actual physical maleness and femaleness vary in time. Of course people can be literal hermaphrodites, or chemically party male and part female at once.
In pre-industrial societies, these factors aren't important; people can do or act how they like, and treat others how they like. Now, in our society of categorisation and organisation, gender issues can cause problems. If equality is desired the solution is as simple: ignore gender.
In this world, all genders mix and gender is not treated as important. Categorisation by gender is illegal. Companies would not be permitted to ask people's gender and governments must not compile or keep any statistics on gender, including for example on birth certificates and medical records, as well as in more trivial areas such as tax forms.
The current statistics on, for example, the gender pay gap, necessarily increase gender division. As well as a gender pay gap, for example, there is a height pay gap; taller people earn more than shorter people, yet statistics on this are more rarely compiled and not highlighted on the news, so height-ism is far less of an issue than sexism. Height-ism is less of an issue not because it is less important, but because the statistics are not tracked and the differences not focused upon. It is by such analysis that divisions occur and are exacerbated. Having a minister for gender equality will, by this very act, create greater gender division by focusing on gender differences. If we had a Minister for Height Equality and compiled and analysed statistics on height then heightism (see, I've already made it one word) would become an issue as important as gender; all conjured up by merely looking at the statistics. By the same token, reversing this trend would naturally create greater equality because it would not be an issue; people really wouldn't imagine or consider gender differences as existing at all. The same, of course, applies to race.
All toilets would be mixed gender, as would all clothes shops and clothing departments in shops. Clothes sizes would need to be standardised.
All sports are mixed gender and divided by ability. It would be quite usual for a football team to be mixed. Each sport would have to carefully define its categorisation to make matches even between competitors; disability sports does this now to a good extent. The differences between Paralympic athletes are usually greater than the gender differences in 'able bodied' athletes.
Of course, socially, people would be well aware of gender, but this system would remove any legal or systematic prejudices.
This is perhaps the most extreme example. It might be, for example, that keeping gender on medical records would help doctors treat a patient (keeping ones entire genetic structure might help too - although that is a more complex and nuanced set of data and would probably need a computer to utilise). Perhaps, in this case, birth gender could be treated as medically confidential.
Perhaps greater gender mixing would increase the risk of sexual violence, in toilets for example, although perhaps greater gender mixing would reduce the risk by normalising gender mixing. In gender segregated systems, single gender schools for example, do people have better or worse understanding of gender differences?
Religion would be another area of contention, with some religions having dogmatic rules regarding gender.