Trade is the language that everyone knows, and is pretty good anywhere, including Yamatai. Almost everybody speaks Trade in the setting.
Yamataian is also spoken in many places, but it's centered on planet Yamatai. Almost everybody on Yamatai (who is not an immigrant) speaks Yamataian. Historically this has to do with the plague killing most of the people on the planet back in YE 38 except for the Geshrin who overwhelmingly spoke in Yamataian (Japanese) because PNUgen was a group of anime-style mad scientists making catgirl clones (also typical of anime!). The further you go from Yamatai, the less people are using it.
Star Army of Yamatai personnel commonly, but not always, speak Yamataian, or at least key parts of it (a Star Army of Yamatai would know everything on the
Star Army glossary even if they couldn't speak Yamataigo fluently). But when there's language barriers people tend to switch to Trade, which everybody knows. An example might be a Phodian or Kodian who might speak Trade and languages of their homeworld, but only a little Yamataian.
The actual requirement these days for soldiers who join the Star Army is fluency in "
Yamataigo (ιͺ馬ε°θͺ) and/or
Trade (language)" which means you can get by with only knowing one of them; however, characters are encouraged to learn both of them as their military career advances. In fact going to a language school might be a good excuse for characters who have absences to explain due to players being unavailable for a while.