Belonging to God’s Household, Christ’s Body, the Church, etc.
In order for a person to be a member of the body of Christ through the New Covenant, (i.e. to be a Christian) there must be a Divine, miraculous change in the heart of such a person. That change culminates in faith and repentance; faith in the finished work of Christ while turning from sin–calling on God to bring salvation.
That is a heart-work no human being can do, but only the Spirit of God. In reading John 9 and 10 (Jesus healing the blind man and Jesus’ subsequent discussion of the “sheep fold,” the “flock” and “other sheep” outside “the fold”), we see how sheep from Israel (“fold”) and “other sheep” (outside of Israel, i.e. Gentiles) are combined together into a “flock” who have one true Shepherd. BTW, this will not be obvious in the KJV (there is a mistranslation of some of the original language).
What is the fellowship or commonality these in the “flock” hold? Is it their outward appearance, genetic background, or anything visible? No, it is a spiritual, Divine fellowship–”my sheep hear my voice and follow Me” and “I know My sheep and they know Me.” This is the essence of belonging to the New Covenant people (to bearing the name “Christian”)–an invisible, Divine spiritual work through which His people follow and know to follow their Shepherd. They trust in Him for forgiveness, salvation and life, and through the Spirit’s power and Christian graces, they keep following until He returns.
Any togetherness beyond this divinely-produced, invisible unity, or any claimed connection to the regenerate (born-again) Body of Christ through heritage, genetics, familial ties, etc. is something other than Christian unity, however pleasant, sentimentally-comforting, etc. that “unity” may be.