Are there any animal species that have two sexes and a kind of different internal diversification in two genders or more? I don't mean something like different-task-based diversity, but something biological? For example let's imagine a species with:
and individuals can match iff they are not only of different sex but also of different gender. So you can only have Ma-Fb, Mb-Fa.
Many organisms have multiple sexes. For instance, here is one which apparently has 7 (and each sex, or "mating type" can only mate with the others). The "gotcha" is that, mostly, they're unicellular organisms where mating in itself is weird to begin with. I haven't ever heard of a metazoan which truly has more than one sex (there may be variation within the males or females, but I think the sperms or eggs are always interchangeable).
Gender, I think, is more of a social term relevant to human behavior. Especially nowadays, so much of the literature and understanding of what is referred to as "gender" is closely linked to things such as gender identity, sexuality, gender roles and personality that is difficult to see how you could meaningfully speak about the "gender" of (non-human) animals.
It's analogous to the idea of "animal language" - certainly some animals may be said to behave in stereotypical ways which influence their mating behavior. Does that mean that what is written about human gender applies? I suspect precious little of it would. I would be quite skeptical of someone discussing gender (in the modern sense) for even uncivilized humans, let alone non-human animals.
Yes, it is called Alternative Mating Strategies and is well known. As an example High-backed pygmy swordtail (a fish) are known for having two types of males, one that is large and aggressive and another that is small and looks like a female to avoid aggression from other males.
I'm not sure what you mean by "gender" in the context of non-human organisms. Gender is usually defined as the intersection between biologically determined sex, and the surrounding culture.
Maybe you could argue that in social insects like, say, ants, the fertile queen and the infertile female workers diverge so much in behavior and gene expression that they aren't the same "sex", or that they could be considered different "genders", but I think that is really stretching the usual definitions of those terms.
The closest thing you will find to what you're looking for, I think, are animals with multiple different mating strategies where one sex (usually the male) can have multiple distinct phenotypes. Some examples include:
Patas monkeys, where some males are large and have harems of females while others are dubbed "sneak maters" and instead of having their own harem they sneak in and mate with females in another male's harem
Scarab beetles, where males either have a horn or lack one. Horned males defend their females at their tunnel entrances while hornless males sneak into tunnels to mate with females
There are hermaphrodites and gender changing (M->F, F->M) fish. Also occasionally, there are also individuals who are neither male nor female, which may or may not count here.
However, I haven't seen 3+ separate genders except for fungi, which might occasionally be multicellular, but probably not what you are thinking of. Fungal genders are so many because they define combinations where mating is compatible. This news story has a decent summary of why as far as gametes goes, larger animals are known to have only 2 gametic genders, while fungi can have thousands. You might say that bacteria have gender when they want it (when they have a pili-containing plasmid) and otherwise they have none - any individual may transfer DNA to another under the right conditions (like death).
I know this isn't what you are asking about, but this is the closest I can find. Digging around, biologist Joan Roughgarden has written a book "Evolution's Rainbow" about cases where there may be more than one social role in a given species for biologically male or females - dominance, display, homosexuality and behaviors like this. Social gender roles has been a way of talking about the biological nature of human social roles since the 1970s - see also Sarah Hrdy.
Tetrahymena thermophila is an organism with 7 different "fuzzy" genders, not only male and female. Its seven sexes are rather prosaically named I, II, III, IV, V, VI and VII. An individual of a given sex can mate with individuals of any except its own, so there are 21 possible orientations. Here is the link: Zoologger: The hairy beast with seven fuzzy sexes
The white-throated sparrow has two sexes (male or female), but also has an addition "gender" (tan or stripe).
So male will only mate with female. AND Tan will only mate with Stripe
This behavior is probably due to a massive chromosome inversion of 1100 genes on chromosome 2. It is starting to cause the birds to act as if it were a second sex chromosome.
http://www.nature.com/news/the-sparrow-with-four-sexes-1.21018
The males of some Lake Tanganika cichlid species such as Lamprologus callipterus exhibit extreme dimorphism of behaviour and size (see e.g. https://www.jstor.org/stable/40407772). These represent alternative reproductive strategies for mating with the same female population though; the big aggressive 'nest males' attempt to guard a 'harem', while the small 'dwarf males' attempt to sneak in, exploiting the fact that they look similar to females and can sometimes deceive the nest males.
I'm not sure this is completely answers your question though. The assortative "Ma-Fb, Mb-Fa" example you give almost certainly does not occur (at least for long) since that would result in two reproductively isolated populations which would speciate rapidly.
