"Jenny is not a girl but my cat."
Translation:Дженни — это не девушка, а моя кошка.
I'm not exactly sure of this, but I think that не negates the invisible verb "is", and doesn't negate the word девушка/девочка. Why that is the case is completely unclear here, since девушка/девочка both mean a human girl, while кошка is a feline girl, so negating the fact the Jenny is not a human girl is not illogical or nonsensical.
Not sure if you know, but девочка refers to a young girl - a child, whereas девушка refers to a girl from about the age of puberty on into womanhood.
The issue with the genitive here, is that genitive would be used to say there isn't any of something, but if you are saying someone or something isn't something else, then it remains in nominative case.
Just found this and now I understand: https://www.duolingo.com/comment/11536858/A-Guide-to-Using-%D0%AD%D0%A2%D0%9E