We don't move from our physical family directly to our Christian faimily when we become Christians. We make God our Father, our source and sustainer of our lives, and the One we are to behave like, then, consequently, we become brothers and sisters of others who have done likewise. This distinction is significant. In fact, this is what we have in common, that we are His spiritual children. And if at any point we begin to value our connection to our brethren more than our connection to our heavenly Father, we have begun to serve an idol instead of God and we have lost that very thing that connected us.
Many Biblical passages that are pressed into service of the Christian unity message are not teaching anything of the kind. It would seem as though those who view the church according to fleshly, worldly principles, who view it as a human institution rather than a spiritual body, being given its life and being maintained by God Himself, think it is their duty to maintain church attendance numbers by promoting unity among believers. This is a mistake. Christian unity is only valuable insofar as it is truly Christian. Our unity in Christ will extend as far as we are walking in the truth and love of Christ, and no further.
One such example of a Biblical passage being misread to support the unity message is John 17. It is often referred to as "Jesus' prayer for unity." But the references to unity here are ambiguous in themselves. These references, "that they may be one even as We" (v. 11), "that they may all be one" (v. 21), "that they may be one" (v. 22), and "that they may be perfected in unity" (v. 23), could be understood to mean "being united" or "being one" with each other, or they could be understood to mean "being united" or "being one" with God. The context must determine the meaning here.
What am I getting at here? Is unity among believers bad? Would I like it if Christians were all fighting with each othere? Certainly not. But God does not want unity for unity's sake. When unbelievers unite in their evil persuits it does not please God. When the people, in Genesis 11, united to build a city, and a tower, and to make a name for themselves as a unified group, God confused their language and scattered them abroad over the face of the whole earth.
It is a question of priorities and pragmatism. Priorities because being united with God in the truth and in love is infinitely more important than being united with people.
Pragmatism because persuing unity with God will bring about the proper kind of unity with other believers, while persuing unity with other believers will not succeed, and it may comprimise our unity with God in the process. I said, "the proper kind of unity" because following God's truth and making ourselves holy in love for God and others will cause sinners to hate us. This may include people in the institutional church—those attending our church congregation with us! We may have to make a choice between unity with God and unity with men. But there is no value to unity at the expense of holiness. If we choose to maintain unity in a group at the expense of the truth and love, we no longer have a fellowship of the love of God in Jesus; we have a fellowship of the desires of man.