Mixed sex chimeras are possible in mammals, and could be described as both male and female, but not a third sex. But that's very much an edge case, and really not part of the transgender debate.
Mixed sex chimeras are possible, but in 99.9999999999999999.... % of cases they can still be described as either male or female, not both.
I hold out that incredibly rare 0.000......1% possibility, as it's hypothetically possible for a mammal to be born with compete and even chimerism of the reproductive organs.
Note I say hypothetically possible, as it has never been documented.
The closest we have is one single case of a human that had the chimeric form of ovotestisticular disorder, had a male reproductive system, and had fathered children with their fertile sperm from one testicle, and whose other scrotal organ was comprised of ovarian tissue... and which on post-mortem examination suggested that perhaps at one point in their past, they had released an egg from that ovary.
But given that this individual had a complete male reproductive system AND had the added benefit of being proven fertile, this individual would be classed as a male. The existence of an ovary alongside their single testicle in their scrotum doesn't make them not male. It doesn't make them both.
Some disorders of sexual development are more common in other species, because other species have different methods of sex
determination. Chimerism is more generally more common in birds than in mammals, due to birds being WZ with the female carrying the mixed pair. Chimerism is also more common in animals (both mammals and birds) that have multiple embryos during gestation (litters or clutches). It's also more common in animals that are artificially inseminated - including livestock. Freemartin cattle are chimeras. Chimerism occurs in a few different ways. It can happen when the egg begins division prior to being fertilized, and ends up fertilized by more than one sperm - this is more common in birds and has something to do with how eggs develop but I'm shaky on the details here. It can also happen when a single, non-divided egg gets fertilized by more than one sperm, in which case the zygote is comprised of a mix of cells from each sperm's haploid. The other way it can occur is when two different eggs are each separately fertilized, then those two separate zygotes fuse.
IIRC, that last type is where we get conjoined twins, because the zygotes are only partially fused. It's also how we get freemartin cattle, because two different sexed fetuses share a fused umbilicus, and the female twin gets partially masculinized because it receives blood that has been shared with the male, and which contains the male's masculinizing hormones. The freemartin remains female, however.
Anyway, that's a lot of unnecessary words to say that chimerism occurs, including mixed-sex chimerism. It's more common in birds than in mammals. A mixed sex chimera is still almost always going to be one sex or another - because sex isn't determined by what characteristics the individual displays but rather by the reproductive anatomy that it develops during gestation.
Mosaicism is something different, and I don't fully understand it. It's a lot more common overall, but rarely causes any deleterious conditions. Mosaicism is responsible for tortoiseshell and calico cats. That's about where my knowledge runs out

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