Im going to try to write what i think it would be like to have a third sex.
thru meiosis, the cell division process that eggs and sperm go through, the chromosome number halves from 46 to 23. the process of contraception unites an egg and sperm together to form a fertilized egg with 46 chromosomes again.
to have a third gender/sex, there must be three types of germ cells (egg/sperm) of which any two can unite to form a fertilized egg. Reproduction for multicellular organisms have only been combinations of egg and sperm. And since there have not been a third type of germ cell, there will not be a third sex.
If however, by some freak mutation, we have three types of germ cells then we would have a third sex/gender.
This third sex would have equal status among male/females. Lets call it a non-male as an example. Since any two of throse three genders are capable of reproduction, we would have males being attracted to non-males and females, non-males being attracted to both male and female, females being attracted to both non-male and male.
Society would be pretty much the same, except we would have three bathrooms.
Bacteria on the other hand, are not only capable of self replicating, but also have the ability to pair up with each other to swap chromosomes. By this mechanism, bacteria are hermaphrodites.