Romans 16:16 (NKJV)

16 Greet one another with a holy kiss. The churches of Christ greet you.

Why is the church called the church of Christ? First, the church of Christ is not a proper name nor is it a title we wear. "church of Christ" is a possession of the risen Lord Jesus Christ. The apostle Paul told the christian men in Ephesus to Love their wives like Christ loved the church and  gave Himself for her, that He might sanctify and cleanse her with the washing of water by the word, that He might present her to Himself a glorious church, not having spot or wrinkle or any such thing, but that she should be holy and without blemish. So husbands ought to love their own wives as their own bodies; he who loves his wife loves himself. For no one ever hated his own flesh but nourishes and cherishes it, just as the lord does the church. For we are members  of His body, of His flesh and of His bones. "For this reason a man shall leave his father and mother and be joined to his wife and the two shall become one flesh." This is a great mystery, but i speak concerning Christ and the church. Ephesians 5:25-32. 

Secondly, we see in scripture the namelessness of Christ's church, ie, "To the saints who are in Ephesus, and faithful in Christ Jesus." Ephesians 1:1; Paul, an apostle (not from men nor through man, but through Jesus Christ and God the Father who raised Him from the dead), and all the brethren who are with me, To the churches of Galatia: Grace and peace from God the Father and our Lord Jesus Christ. Galatians 1:1-3; Paul an apostle of Jesus Christ by the will of God, and Timothy our brother, To the churches of Christ which is at Corinth, with all the saints who are in all Achaia. 1 Corinthians 1:1; Paul and Timothy bondservants of Jesus Christ, To all the saints in Christ Jesus who are in Philippi, with the overseers and deacons: Grace to you and peace from God our Father and the Lord Jesus Christ, Philippians 1:1-2;  Paul an apostle of Jesus Christ by the will of God and Timothy our brother, To the saints and faithful brethren in Christ who are in Colosse: Grace and peace from God the Father and the Lord Jesus Christ praying always for you, Colossians 1:1-2. These are but a few of the many greetings to the churches of God and of the Lord Jesus Christ found in scripture.

Thirdly, We Christians bear the name of Jesus Christ our Lord, this is why we are called, simply, Christian. We are His possession, He bought us back from the slavery of sin, He sanctified us and cleansed us by His blood, He redeemed us from sin. That means he bought us, and we are His. When we submitted to Jesus' Lordship, we did it willingly and without force, and we are His bondservants. We volunteered to submit to the risen Lord Jesus. Ephesians 2:11-16, 1 Corinthians 15:1-5, 1 Corinthians 15:20-28, Ephesians 2:1-6.