Some individuals produce both.
No mammal produces both. No mammal, no human, no "individual" within the context of this discussion. Humans are not clownfish.
Which doesn't make them something other than male or female, it just makes them infertile males or females.
Two gametes implies four categories, not two.
No, it doesn't. If you disagree, take it up with evolutionary biologists. Infertile males are still males. Infertile females are still females. And at no point does the question of fertility or its absence switch someone to the opposite sex, nor does it make them a new third sex, nor does it make them unsexed.
There are two ABO antigens on red blood corpuscles; Giving four blood types - A, B, AB, and O*
This is irrelevant. Three primary subatomic particles, which give rise to 118 atoms. This is just as irrelevant as your mention of antigens.
Two gametes implies four sexes (if we accept the unstated assumption that gamete production is the definitive factor in determination of sex, which is a deeply flawed assumption, but that's a whole other discussion).
To get two sexes, there would need to be only one gamete, which individuals either produce, or don't.
No, you are wrong. I mean, just completely and irrevocably wrong.
Let's stick with the basic failure in your assumption. For your assumption to work, we would need sex-related chromosomes to be codominant as well as having the ability to be absent completely. But that's not the case.
Each parent contributes ONE sex chromosome. Female parents can only contribute an X, they have no Y to pass on to their offspring. Male parents can contribute either an X or a Y. Barring karyotypic anomalies, offspring receive an X from their mother and either an X or a Y from their father.
If the child receives two X chromosomes, they are female. If they receive an X and a Y, they are male. Those are the combinations, and the resulting sexes. This is important, because the triggers that drive the mullerian process are on the X chromosome, and the ones that turn that process off and instigate the wolfian process are on the Y chromosome - specifically, the SRY gene.
If something goes wrong, and the offspring fails to receive an X chromosome from the mother, the fetus is inviable and dies. If the offspring receives an X from the mother, and something goes wrong and they don't receive either chromosome from the father, it is possible for the child to develop as a single-X female.
Even with all of the various karyotype combinations that can occur when something goes off the rails... if the offspring receives a Y chromosome from the father, it will be a male. The Y chromosome is dominant.
There are two semi-exceptions to this: Swyer syndrome and CAIS.. In the case of Swyer syndrome, the offspring inherits a damaged Y chromosome from the father - a chromosome in which the SRY gene doesn't work. Thus, the fetus never receives the prompt to switch from mullerian to wolfian developmental paths. People with Swyer syndrome are female phenotypical, but are sterile. They develop female-typical fallopian tubes, uterus, cervix, etc. but they don't have ovaries, they have undifferentiated tissue in place of ovaries. In CAIS, the SRY gene is there on the Y chromosome, but the body's ability to receive signals from it is damaged or absent. The receptor is carried on the X chromosome. For Complete Androgen Insensitivity Syndrome, the fetus develops female-typical anatomy, but generally has testicular tissue in place of ovaries. Most people, including doctors in any fields other than reproductive medicine, both of these types of people are treated as female. Note that Partial Androgen Insensitivity Syndrome produces people who are fully categorized as male, even if they have some underdeveloped reproductive features.
Please don't pass along false pseudo-science. Chromosomal variations aren't new sexes. There are only two sexes: Female, and Male.