Yes, the noun 'family' is a common noun, a word for any related
group of people or things.
A proper noun is the name of a specific person, place, thing, or
a title; for example:
Jill E. Family, Associate Professor of Law, Widener University
School of Law, Harrisburg, PA
Family Street, Fayetteville, NC or Family Circle, San Diego,
CA
Family Dollar Stores, Inc.
"Family Business", 1989 move with Sean Connery and Dustin
Hoffman