Generican Empire: Well, the thing to remember here is that the Empire of the Star was named long before it ever got into space (and has many stars, besides). It’s actually named after a philosophical symbol rather than anything physical. (In fairness and trope-compliance, they could have called it the Eldraeic Imperium, or some such, except (a) back on their own world at the beginning they didn’t feel like offering up such a blatant statement of world-conquering intent, and (b) once they got out among the stars, they didn’t want to sound like a bunch of race/species-obsessed jerks.)
By the time it acquired the alternate soubriquet of the Eldraeic Transcend, evidently people figured they’d already established their credentials on the polyspecificity front.
(The third alternate name, the “Bright Empire”, is actually a given-in-scorn-adopted-in-pride name originating with a news editor’s slip of the tongue and an unfortunate phrasing in an interstellar explorer’s autobiography; as such, in the Imperial mouth, it can make you sound isolationist, hegemonic, or even belletrist, and in the outworlders’, either distinctly pro-Imperial or distinctly sarcastic, and should thus be used with caution.)
Anyway, back on topic: most star nations hang some sort of non-generican name on themselves. The ones which don’t are often considered suspicious, maybe even likely to be rogue – sort of the way that, on Earth, any country calling itself a Democratic People’s Republic is definitely some sort of creepy-ass dictatorship.
Averted somewhat by the “Associated Worlds”, which isn’t all that creepy – it’s just that a name that Generican is all that the various people included under that moniker could agree on. They are all worlds (well, mostly, but the drifts that disagreed didn’t push that point too hard) and they are loosely associated, at least galactographically. Everything else may vary.