I think that "Caldanese" as a term might be influenced by Battletech's Swedenese (Rasalhagian Swedish-Japanese pidgin-turned-language). Of course, it likely loans a whole lot from Gallentean, though likely has been a target for national romantic rebuilding. I've used the rather utilitarian term "Standard Caldari (Language)" for the language. (I think Napanii would be akin to some kind of Deteis-Latin within the extended fiction.)
The Amarrian language, I think, is mostly called Amarrish, though an empire of that size would likely have several, with the Imperial language being the official one with other languages being regional ones. India might be an example here, with Hindi and English being common languages and then there being a number of other languages. There might be several languages used only for historical and religious research (compare to, say, Latin, Aramean, Ancient Greek...). One of these might even be the official Imperial language, even if not a word is seen outside Dam-Torsad and the only use would be giving out edicts and writing laws (with the text being immediately translated to modern Amarrish before being distributed).
Gallentean have, of course, Gallentean. And languages for each nationality, but Gallentean is likely the lingua franca (pun intended).
I don't really know about the Matari RP, but I think we had Word of God that most Minmatar can speak Amarrish, even if they might use it only after all other attempts at communicating have failed. I would expect that there's been a significant effort to revive old tribal languages similar to Hebrew revival.