In-game, all of the characters - from Noctis and the Bros in Lucis, through Altissia and Niflheim, plus the refugees from Galahd - speak the same language.
There’s no indication that the different nations have (or had in the past) different languages, but I still like thinking about the concept.
Which makes me wonder what the universal tongue of Eos would be.
It would be some variant of either Niflheim and Lucian, if I had to guess - since both are the only ‘conquering’ nations over a large amount of land (that we know of), and it’s the conflict between them that is the primary setting of the game (until the prophecy kicks in, but still).
Niflheim is the modern conquering empire of the world of Eos, controlling the majority of (known) countries over at least one continent and an island state, and the majority of Eos uses their currency - Gil.
So, the common tongue possibly was rooted in the language of Niflheim.
But it’s also implied that Lucis was once a conquering empire. In the Comrades dlc, there was that one purchasable costumization (I think Body Art Anthology? It was one of the tattoo design purchase for the character from the stores) that referenced Cleigne or another region and referred to the art of that area’s indigenous people - if I’m remembering right.
(If anyone can check this please do so, while I do have the game, I can't play comrades on my laptop - does not load).
As in, Lucis didn’t originally have that region fall under their borders, and people had resided in the area before then - and then one of the previous kings of yore (or regents, who knows?) had that land conquered.
(In-game, Lucis feels so small - you can drive across the whole country in the car. Which makes it kind of easy to forget that the kingdom of Lucis, as a whole, not counting the wall which essentially segregates the Crown City from the other regions, is literally the entire continent.)
So there’s the potential that the common language, though it could be called Niflheim, is already related to common modern Lucian. My reason for thinking about this:
At least one King of Lucis also has their tomb on their other Niflheim/Tenebraen continent - which can mean either it was from a period of time the two nations are friendly as is the Empire’s official stance or Lucis also controlled some land on the other continents as well.
Additionally, We know that that Nox Fleuret family which established Tenebrae originated in the same region as Modern Lucis/Insomnia since the First Oracle, Aera Mirus Fleuret, worked along the founder of the Lucis Caelum bloodline.
When they left the Insomnia region (presumably after Aera’s death), others likely left with them - and maybe that group split off and would eventually become the Aldercapt Dynasty of Niflheim.
Meaning they also originated in the pre-lucis period as Lucians, probably spoke the ancient precursor to the modern Lucian language (which is the equivalent of IRL latin, seen by the naming conventions of the Lucis Caelum bloodline and other characters).
Even as the languages changed overtime, they would still originate from the same place - the Pre-Lucis Insomnia (and possibly Leide?) regions.