It does no violence to the Scriptural record to propose that God exists in the nature and substance of family.
The Bible states plainly that "there is one God" (I Timothy 2:5). It also teaches that three distinct personalities exist in the oneness of Divinity.
"Thou art my Father, my God, and the rock of my salvation" (Psalm 89:26).
"Unto the Son He saith, Thy throne, o God, is forever and ever" (Hebrews 1:8).
"But Peter said, Ananias, why hath Satan filled thine heart to lie to the Holy Spirit?... Thou hast not lied unto men, but unto God" (Acts 2:4-5).
God exists as Father, Son, and Holy Spirit, three sublime personalities so intrinsically united in nature, character, and being that "They" are one. A plurality in unity, a unity in plurality, and herein lies the reference to family. Our own experience tells us that distinct persons can be so bound in heart that oneness is easily the best way to describe the loving union. The love of husband and wife, parents and children, and sibling with sibling united in the bond of family reveals that in God's creation, unity and plurality are mutually inclusive rather than exclusive.
We need little understanding or explanation for such glory. Some realities are best known in that light of the heart so sublimely beautiful that no description is possible. I recall an evening with our family many years ago, when our children were still young and all lived at home. We were gathered around a table at a favorite restaurant, and the banter was lively. For a brief moment, I sat back to gaze upon and listen to the four people so dear to my heart. I cannot explain or describe that moment, but deep within my depths, I knew then (and I still know) that God gave me a glimpse of something so beautiful that tears streamed down my face in the blessed moment (as they do now in recollection). The "something" concerned my own family, no doubt. I realized the amazing gift I had been given. However, I also believe that the glimpse involved more. I think it involved God Himself, and a hint at the wonder of His triune heart and being.
I came away from the moment believing that God can be defined, in His essence, in terms of family. He is one and yet He is three; He is three and yet He is one. The Bible unapologetically proclaims the enigma, and the God of the Bible provides the most definitive and influential aspect of our lives - family - as a window into who and what He is. Indeed, family is elemental in the existence of humanity because humanity's Maker exists in the same wonderful reality. Yes, in essence, God the Father, God the Son, and God the Holy Spirit comprise a family.
Amazingly, born again believers in the Lord Jesus are now part of "the whole family in Heaven and earth" (Ephesians 3:15). We are adopted sons and daughters, of course, and will never become God as He exists in His essence. However, we are so spiritually united to the Lord Jesus that we are "in Christ," our Heavenly Father having drawn us as near to Himself as created beings can be. Our Savior prayed for our entree into such glory, and then died and rose again to make Heaven our eternal home not only in place, but in personal kinship to the Father, the Son, and the Holy Spirit...
"Neither pray I for these alone, but for them also which shall believe on Me through their word; that they all may be one; as Thou, Father, art in Me, and I in Thee, that they also may be one in us."