This post is in need of some updates, but the short answer is that yes, each of the Spoken races has (or has had in the past) it's own unique language. Some of those languages are never used anymore and have all but died out, some are still spoken just not Eorzea, while still others get used but we don't get to hear them really due to our ability with the Echo translating all languages for us.

Languages of Hydaelyn Lore