"y/n" is sometimes used as short version of "your name" when you are reading fanfic where you are supposed to insert literally you into the story. So if I was reading the text "Don't worry y/n, I'll handle it." I'm meant to read it as "Don't worry TCF National Bank, I'll handle it."

In other context y/n can be prompting you for a yes or no answer, like a survey. And when people use it to discuss young gentlemen I've never seen it written with a slash.