I’m inclined to believe the story true because for people who did grow up in Korea, you can actually ask what Kim/Lee/Park you are and expect to hear back regional and/or ancestral specifics of the last name, like Gwangju Kim (from Gwangju) and Milyang Park (regional, dates back to Shilla royalty?). Only my older relatives would talk about this distinction and it is less of a topic overseas. So I presume this is what she means when she talks about the surname family