BE family is a well-established idiom meaning "to be (or to be accepted as) a member of [the speaker's] family".
I suspect that it arose in the first instance because English has no adjective which means "belonging to a family". The regular adjectival form of family, familiar, very early took the figurative meaning "intimate" as its primary sense; consequently the noun itself, used attributively (family jewels, family tradition), has been since the 17th century the only form available as an adjective for the literal sense.
However, use as a predicate adjective (e.g., "They're family") was largely colloquial and had a "down-home", almost dialect feel until the hippie era, when the notion of unrelated individuals constituting a "family" took hold. Sister Sledge's enormously popular 1979 song, "We Are Family", became the anthem of this sentiment (though the song itself celebrated an actual family relationship). Since then the expression has flourished, as may be seen in this Google Ngram: