I agree that God is not a family by Himself or multiple persons. However, the scriptures do mention the "whole family in heaven and earth" (Ephesians3:15) but I think this means God and His children. The saints.
Biblical names for the church..
Assembly of the saints Psalm 89:7.
Assembly of the upright Psalm 111:1.
Body of Christ Ephesians 1:22,23; Colossians 1:24.
Branch of God's planting Isaiah 60:21.
Bride of Christ Revelation 21:9.
Church of God Acts 20:28.
Church...