Fae don’t play by the same rules as mortals. Just because they possess your name doesn’t make it their name. It just makes it not yours. And Fae magic doesn’t necessarily rely on truenames, it relies more on expression and emotion. If you think that your name is your name, and now a Fae has taken it from you, it gives them power over you. And most especially Fae power is held in deals, contracts, favors, tit-for-tat. They asked you for your name and you gave it freely. That’s a contract. If you want it back, you’re going to need to draft a new contract.
Besides, most fae won’t exert such complete control. Some will, definitely. But most of them prefer to trap you inside dealings of your own design. You’ll make an initial deal, which you know is a bad idea but you don’t have much of a choice - it always starts like this, with an offer you can’t refuse, such as recovering your name after giving it. Then you make another deal to escape the consequences of the first one. Then another. Then another. Before you know it you’re bound by multiple conflicting deals and can’t possibly keep your word on all of them - and then that’s when they really get you. Broken promises hold terrifying power with the Fae.
Unironically I keep a devil on retainer for this. I’ve interacted with the Fae twice and it went poorly both times. I’m not speaking to them again without my lawyer present.