It's unfortunate that
Fanny is slang. I guess it's technically usable as a nickname, but who'd want to be called it? No one I know. As a full name, I don't think so. It'd seem incredible.
I'd like it better spelled
Fanni or at least
Fannie.
I think it's for
Frances.
I think
Dick seems not as bad because of being more common and still borne by well known men, so people can picture a man, rather than it just being a word hypothetically used as a name.
I don't think Dicks get teased because they're all adults and it'd be too juvenile to tease them to their face.
I doubt anyone would accept it as their nickname now, though, unless it was kind of a joke, like when a guy gets a 'bad' nickname from his friends' raillery.
- mirfak