Fey Names

Had an idea with the Fey, curious if it could feasibly work.

If a Fey creature asks you for your name (with the intent to literally take it) what happens if you give them someone else’s name? Like if I had a character named Mark who knew someone named John, if a Fey asks Mark for his name and he says John, does John lose his name? Or does the Fey just take the name and Mark keeps his name because he gave them a fake name.