I know it makes grammatical sense, but it's still hard to parse a sentence that doesn't make any logical sense... is there a particular type of elephants that is supposed to pertain to a woman as opposed to a girl? Obviously, I would need to know so that I can make sure I get the right type of elephants. It would be very embarrassing. This is very important.
I could be wrong but I think in this case there isn't much difference between the use of the definite and indefinite articles. It could just as easily be "En pige har kvindens elefanter" and it would still mean the same thing. So it doesn't mean that there is a certain type of elephant that is "of women," it means that a woman owns elephants but the girl has them. I might be wrong though, I'm not a native speaker.
No, it has correct grammar. This is because, the woman has multiple elephants. Therefore, the girl has a woman's elephants. If "woman's" was excluded you'd be right. It would be, the girl has an elephant, the girl has elephants, or the girl has some elephants. But since the elephants belong to the woman, they are not referred back to the girl.
"En pige har en kvindes elefanter." / "The girl has a woman's elephants."
The meaning of the sentence is that "A girl has the elephants" that belong to "a woman".
In Danish, if the girl had an elephant that was female it would be: "En pige har en hunelefant." ("Hunelefant" = An elephant that is a she")