As we said, it's totally up to you. Zuko's face is pretty recognizable, so I guess it would depend on if you think he'd done anything to get noticed attached to his face! If you don't want him getting his name taken, you can just err on the safe side and have him keep his helmet. However, if you're looking for a reason to get him into some trouble, you could easily just say that if he's not infamous or famous for anything, the fae have their own shit to deal with and would be like "I swear I've seen his face before but who knows from where..."

Characters aren't informed unless they ask (any of the fairies of the court will tell if asked), but a conversation with a few people should make it easy enough to figure out.