I'd say the majority of the language is based on the Allagan Empire's language, considering they controlled the majority of all 3 continents so even though they were destroyed in the Fourth Umbral Era. I would say the current language is based on it or even languages before it. (Hello Deidre).

You have the Sylphs, Moogle speak, and the Beast Tribes language. The Beast Tribes are able to understand each other easily and many people can communicate and trade with the Beast Tribes so either it's a different language and easily learnable or they're just using our language, I'm not sure if it's said. But on the other hand, not everyone can understand the Sylphs or Moogles.