"Fairy" is a very broad term that has been used to describe everything from ghosts and various spirits to even demons and other creatures. It's about as broad a term as "Dragon," and thus it makes sense, seeing as Fairy might arguably be a counterpart to Dragon.
Plus, we already have a Fairy egg group. There is no Light egg group, nor has anything in Pokemon ever been associated with the word "Light." (Moves that revolve around literal light are usually given solar-related terms.)
True. Every single localization of Sylveon refers to fairies, and several contain the word "Fairy". Probably the reason they didn't name it "Faireon" for English localizations was because they knew that a lot of North Americans unfamiliar with most fairy folklore would freak out and claim that the type was "girly" or "gay" (because homophobia isn't wrong, right guize?). Which is probably why the Fairy type will be localized as something like Fae or Sylvan.
When life's really getting you down, take solace in the fact that you're going to die.