Ecolect
According to wiktionary an ecolect is a language variety unique to a household although I might cast the net slightly wider than a single household, maybe to an (extended) family.

However it appears that as yet it has not made it’s way into the OED or many other mainstream dictionaries.
Compare with:
Idiolect: The language variant used by a specific individual.
Ethnolect: A language variety specific to an ethnic group.
Sociolect: The variant of language used by a social group such as a socioeconomic class, ethnic group, age group etc.
Dialect: A variety of a language that is characteristic of a particular area.