Can Fae take your new name?

I've heard a lot of stuff about trans people giving Fae their dead name, and once they do, it's like it never existed, so their prefered name is their name.

But if, after that, a different Fae asks for their name, and they give their prefered name, could it be stolen?

Would it fall under the category of a fake name? (Because it was chosen, not because it's invalid)