So the fae, insofar as anyone knows, basically just started one day, when Oberon, Maeve, and Titania came strolling out of wherever it is they came from and said "Yeah, this'll do." They did say it. The three of them began with a common language, which we can call, for lack of a better word, "Fae." They spoke the language of the fae, and that's not what they called themselves in that tongue, because "fae" is a loan word. They didn't call themselves by name, either. They were the only three things that mattered in the entire world, and when you have a population of three, you don't so much need proper names.
Now, creating a language is hard. There's a reason that most of us are pretty relieved when we discover that hey, there's a word for that. As the Three wandered around, encountering people and making trouble, they began acquiring words for things. Tree. House. Car. Uncivilized behavior. Frog. Witch. Spell. Humans turned out to be incredibly useful in the "naming things" department, and the Three wound up being called things other than "one I'm with" and "one I'm not."
When it became apparent that whoa, hey, all their kids were totally radically different from one another, and so were their children, humanity stepped up to the plate again, slapping all kinds of names on them. (This is why so many fae races have names that translate as either "funny-looking people" or "kinda like a whale/horse/tree/whatever, only not.") The fae, lacking any better ideas, sort of rolled with it. This is why a) so many fae races have names from so many different languages, and b) fae pronunciation and grammar is a little...questionable. They're literally their own messed-up polyglot linguistic drift.
That's where their consistent vocabulary (race names, etc.) comes from. There are regional variations (Kitsune in Japan don't call Firstborn "Firstborn," they have a local name that I can't spell), but for the most part, those pieces will remain consistent. As for conversational "I can talk to you, you can talk to me" speech, that tends to fit whatever the local language happens to be. So Toby speaks mostly English, as do coastal Undersea fae. Li Qin speaks both English and Mandarin. The Luidaeg speaks about eighteen languages fluently, and can tell you to go fuck yourself in any and all of them.
Fae who have been isolated from humanity for any length of time will tend to develop their own language, although the anchored "root words" will remain, as artifacts to facilitate communication with other races. The deep Undersea has its own language, as does the Oversky. Some fae are not equipped to speak human words, and find other forms of language. Dryads are fluent in wind, for example. But when it comes to the spoken word, the fae are thieves, and they don't give a damn about your grammar.