suffix
Suffix attached to surnames to refer to a family or household, often in formal or historical contexts.
徳川家は江戸時代に将軍を出した家系だ。
The Tokugawa family is a lineage that produced shoguns during the Edo period.
あの家は代々医者の家だ。
That family has been a family of doctors for generations.
うち is a casual noun for 'home' or 'one's own family', whereas け is a formal suffix attached to surnames.
The suffix け is derived from the noun いえ (house, family) through regular sound change and is used to indicate a family or household, especially after a surname.