Reddit reviews Homegirls: Language and Cultural Practice Among Latina Youth Gangs

The sociological notion of a community of practice might be useful here. Basically a CoP is A) a relatively stable group of people that B) get together on a relatively regular basis in order to C) do some relatively stable thing together.

It is common since the late nineties in sociolinguistics to study one particular CoPs linguistic particularities with the understanding that language is one dimension through which the CoP will define itself. In practice CoPs studied in sociolinguistics tend to explore interactions of language with gender and racial dynamics, e.g. black women in Appalachia, high school nerd girls, Latina gangs... but in theory a CoP doesn't have to involve power dynamics.

So to answer your question, I think one would talk of a language variety specific to a family as a community of practice (or perhaps more accurately a community of place). There isn't be a specific term for it as far as I know.

https://www.amazon.com/Homegirls-Language-Cultural-Practice-Latina/dp/063123490X

Homegirls by Norma Mendoza-Denton is pretty good, I just read it for a cultural/linguistic anthro class.