It's also a nn for Genevieve. And, back in the day, most Jenny people were actually Jane - my grandmother's friend was always known as Jenny but always signed herself Jane.I used to know a JennyEsther. Nice name. And she's just Jenny!
Sure you can. My grandma goes by Jenny but her full first name is Genevieve. You could also use Geneva, Jennica, Jenna, Jenessa, and Jennette. Or you could just use Jenny by itself.