Not a DND expert but generally the trope with Fey is that their magic uses people's names so If a fey knows your name they can do whatever they want to you.
Depending on the way the fey phrased the request you may either have insulted it, or broken a deal. If it asks to have your name and you answer with "yes, my name is <fake name>." You have lied to it and broken a simple transaction. It may even be justified in using ever more harsh measures to extract your name from you, as it has a legitimate claim on it.
At least that is the way I would play such a situation for the fey. For me the thing is that trying to outsmart them rarely works, because they know netter in which rules they operate than you do.
A answer like "You may not have my name, but you can call me <fake name>. " Should work better in my opinion.
Sorry for the wall of text btw. Just being bored out of my mind rn.