1 in 60,000 women have one X and one Y chromosome. They are still women.
The ratio of women who have an X and a Y chromosome is much higher among actresses and fashion models than in the general population. If you are a heterosexual man, there’s a good chance you’ve lusted after a woman with an X and a Y chromosome at one point or another.
In Asiatic mole voles, both males and females have two X chromosomes. We still don’t know what determines whether an Asiatic mole vole turns out male or female. Perhaps it is a choice.
Birds have two sets of sex chromosomes, but they are different from X and Y chromosomes. Let’s call them W and Z.
Male birds have two Z chromosomes, while female birds have one W and one Z chromosome.
In alligators, sex is not determined by chromosomes. If the egg is incubates above 34°C, the alligator turns into a male. If the egg incubates below 30°C, it turns into a female.
If the alligator egg is incubated between 30°C and 34°C, it could end up being either. We still don’t know what determines which sex the alligator ends up being in this case. Perhaps it is a choice.
In sea turtles, sex is also determined by the temperature of the egg during incubation, except that females are the result of higher temperatures instead of lower ones.
In ants, wasps, and bees, unfertilized eggs give rise to males, while fertilized eggs give rise to females.
There are some crustaceans that are born male but turn female at some point over the course of their lives. For some of species, they are all born male and only turn female if and when they are infected by a bacteria called Wolbachia. Presumably, this is a disease they would not want to cure.
Most reef fish change sex at some point in their lives. Many of them change back and forth multiple times during their lifespan. Some end up managing to be both sexes at the same time (hermaphrodites). The ones who are always the same sex for their entire lives are a small, small minority.
This issue was not addressed by the movie “Finding Nemo”.
Sometimes sex is determined by chromosomes, but there are more than two sexes. Physarum polycephalum has 13 sexes.
The split gill mushroom has 28,000 different sexes.
One final note: The marine flatworm Pseudobiceros bedfordi is a hermaphrodite. It also has two penises. This is how they reproduce: Two Pseudobiceros bedfordi will engage in a sperm war, squirting sperm at each other with their penises, each one trying to impregnate the other while trying not to become impregnated him/her-self.
That isn’t particularly related to the relationship between sex and chromosomes. It’s just cool.
These and other interesting facts about sex and genetics can be found in Power, Sex, Suicide: Mitochondria and the meaning of life, by Nick Lane.