You could and it might work, or it might not. Depends on a lot of factors. Thing is, you are still engaging the fey in word games and try to outsmart them in their own games. Being blunt and telling them No, giving clear and unambiguous answers offers less attack ground for them.

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.

I played an absolute Law-and-Order Paladin once. The rogue allways had to find ways around me, because of course he had to steal whatever was not nailed down. He had lots of fun finding ways to both convince me that he was an upstamding citizen and steal stuff right under my nose.