Each familiar is fluent with a few languages, however, it appears that they're not fluent in speaking that language.
Below is a test using a Worg familiar that is fluent in goblin and animal language, with it displaying a clear pattern of failing language checks when speaking those languages when possessing the familiar.
My character is fluent in none of those languages, for instance.
This doesn't seem intended, as the languages work as intended when using the -a command to speak through the familiar.