Look the people that I have seen talk about this have said that they (some) feel both male and female, some feel like gender fluid and a lot of things, but only because you feel this way or like this specific thing does not mean you need a gender so you can differ from others.
Those genders are male and female. There's really no such thing as transgender. No matter what gender you identify as, no matter how many hormone injections and plastic surgeries you have, you will ALWAYS be the gender you were born as. Facts don't care about your feelings. If you were born a man you are a MAN If you were born a woman you are a WOMAN. Go and be transgender if you like. It is not my place to stop you and you are free to chose how you live your life but I certainly don't have to agree with you. It's fact. Male and female are the ONLY genders. Not another can be added.
Gender was always synonymous with sex until the PC SJW's decided that they needed a sugar-coated way of defining a mental disorder, and most people still use it that way. And yes, transgenderism is a mental disorder, as it includes dysphoria with physical reality and leads to other mental issues such as social anxiety and depression, even in cases of relatively little societal pressure. To this day, things like transableism and species dysphoria are still classified as such. You don't see many people advocating that someone is actually a cow because they identify as one. And we shouldn't be feeding into it, that's not helping.
As for agender, it literally means without gender. You could possibly make a case for a third gender with hermaphrodites, depending whether you'd classify a combination of two things as something separate.
Any claim of "sex and gender are not the same", "gender roles are socially constructed", "gender is psychological" not grounded in the science but in the manner of gender politics or "social science" gone bad.
In the issue of homo sapiens, male and female are the only existing sexes produced by evolution. Hence, the chromosomes of XX and XY. The claim of Intersex as a possibility as the third sex of humanity is nothing more than false claim because intersexes are an example of chimerism. Chimerism was produced by two embryos fused together. It's the birth defect, not the third sex of humanity.
The gender roles like masculine and feminine were developed naturally by males and females of humanity. The natural development of gender roles contributed to testosterone and estrogen. This is irrefutable fact.
Non-binary, agender, GNC and other make-up gender roles are nothing more than the case of gender dysphoria. Remember, social science is pseudoscience and not something to be taken seriously.
Source for Chimerism: http://www.brighthub.com/science/genetics/articles/39124.aspx
Source for gender dysphoria:
Source for sex chromosomes and sex-linked:
Source for the development of gender roles linked to the differences in sex, behavior, testosterone, and estrogen:
Source for why social science being not science: https://www.quora.com/Is-social-science-a-real-science
Without getting into bogus politics about "transgendered" people, I am simply going to say, yes, there are two genders.
You can get into semantics and talk about hermaphrodites being a third gender, but even they are largely identified with one gender or the other, depending upon the features the body exhibits in sexual maturity.
There are only two sexes. But there are myriad genders. Two often people relate to gender, the same understanding of sex.
Sex is what is given to you at birth based on your genitalia. It's not a matter of debate unless you get a sex-change. At birth, you are either man, woman, or possibly a hermaphrodite. That is indisputable.
But it is not so definite or binary with gender. In the issue of gender it is psychological, not a matter of opinion or 'feelz'. There is very little to dispute. There is not a definitive Man or Woman in the human gender identity. There is a Man and a Woman, but there is a whole spectrum in between where people can identify how they wish. Sex is binary, gender isn't
Source: I am a gender non-conformist in theory
I believe that there are only two genders for humans, but there are more than two in other species. (I know your thinking this is cheap, but the question never specified). For example there is a type of fungi that has 36,000 different genders. There is a type of jelly fish that can actually change gender. Also in Latin (maybe this is a little cheap) there are three genders Masculine, Feminine, Neuter.
There are 4: male, female, agender, and hermaphrodite. Though you could say that there are only 3 since a Hermaphrodite is only a male and a female, and thus not a unique gender, or that there are only 2 since agender is the lack o a gender therefore not a gender, but these are all legitimate answers as to what gender you are, so you could say that are are 4 or 2
Sex and gender are not the same thing. Most people who say "yes" to this end up arguing that there are only two *SEXES*, and they're even wrong about that; there are a number of intersex conditions.
In short, sex is physical, and most people are within 2 categories of that, and gender is psychological, and though *most* people also fit within two categories as either men or women, there are plenty of others; non-binary, agender, GNC, and various others.