That makes sense; it's just confusing because the lore book specifically mentioned that the Xaela tribes still speak their own language.
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They might just be referring to how they make ample use of Xaela/Mongolian words in their everyday speech, which might be considered unusual as Hingan/Japanese seems to be the dominant secondary language everywhere else in Othard.
Just as Hingans speak the in-game equivalent of Japanese I have little doubt that Pureblood Garleans speak the in-game equivalent of Latin. It is but a question as to whether the language is taught in-depth to citizens and conscripts of Garlemald. It could be something fairly exclusive - nobody is really taught Latin in the UK if they go to a state school but it's very common for it to be taught at private schools. (You can imagine my puzzlement when I graduated and realised that only a tiny portion of the population actually spoke Latin! Yet I digress...)
Miqo'te got their own as well. All Keeper names (and I guess Seeker names as well) are based on that language, and it's explicitely mentioned that it's really hard for non-Miqo'te to make some of the sounds those names are comprised of, e.g. that 'h' most of them have in them (like, for example, Okhi). Also, the lorebook mentions they took on common quite early on, but still use parts of their own language in, for example, hunting.