The Bible says the Church is Christ’s Body (Col. 1:18)–he’s separating the two. He’s doing this because he believes in a kind of invisible body that is not tightly compacted like the Bible says (Eph. 4:16), but rather is loosely scattered among various visible bodies each professing different things. This is not a Biblical model of the Church.
On the other hand, we are baptized into the one Body in the one Spirit (1 Cor. 12:13), partake of the one bread as one Body (1 Cor. 10:17), profess the One Lord, one faith, one Baptism (Eph 4:4-5). Like Noah’s ark, you have to be on board (as the Bible says, like Noah’s ark in the flood, Baptism now saves (1 Peter 3:20-21), and as I just showed, we are baptized into that one Body–so the ark and the Body have the same significance in this regard).
Furthermore, to have fellowship or communion with Christ, you must have fellowship with those who have fellowship with Him (1 John 1:3). We are forbidden, therefore, from schisms and must be united in belief (1 Cor. 1:10).
Furthermore, while Christ is the head of the Church, He has also willed that the Church be led by men. He put St. Peter in charge of His flock (John 21:15-17)–therefore, to be a member of Christ’s flock is to be a member of the flock tended by Peter. Plus, while bishops are selected by men, they ultimately receive authority from the Holy Spirit (Acts 20:28).
Baptists long ago separated from this one body through schism and breaking the unity of belief. They refused to be part of the flock governed by Peter and the other bishops appointed by the Holy Spirit. They also no longer partake of the one bread as His one body.
As the for the Church and Christ being identified together–they are the same in one sense, but distinct in another. The CCC explains in the following two passages:
[quote=CCC]795 Christ and his Church thus together make up the “whole Christ” (Christus totus). The Church is one with Christ. The saints are acutely aware of this unity:
Let us rejoice then and give thanks that we have become not only Christians, but Christ himself. Do you understand and grasp, brethren, God’s grace toward us? Marvel and rejoice: we have become Christ. For if he is the head, we are the members; he and we together are the whole man. . . . The fullness of Christ then is the head and the members. But what does “head and members” mean? Christ and the Church.230
Our redeemer has shown himself to be one person with the holy Church whom he has taken to himself.231
Head and members form as it were one and the same mystical person.232
A reply of St. Joan of Arc to her judges sums up the faith of the holy doctors and the good sense of the believer: "About Jesus Christ and the Church, I simply know they’re just one thing, and we shouldn’t complicate the matter."233
796 The unity of Christ and the Church, head and members of one Body, also implies the distinction of the two within a personal relationship. This aspect is often expressed by the image of bridegroom and bride. The theme of Christ as Bridegroom of the Church was prepared for by the prophets and announced by John the Baptist.234 The Lord referred to himself as the "bridegroom."235 The Apostle speaks of the whole Church and of each of the faithful, members of his Body, as a bride “betrothed” to Christ the Lord so as to become but one spirit with him.236 The Church is the spotless bride of the spotless Lamb.237 "Christ loved the Church and gave himself up for her, that he might sanctify her."238 He has joined her with himself in an everlasting covenant and never stops caring for her as for his own body:239
This is the whole Christ, head and body, one formed from many . . . whether the head or members speak, it is Christ who speaks. He speaks in his role as the head (ex persona capitis) and in his role as body (ex persona corporis). What does this mean? "The two will become one flesh. This is a great mystery, and I am applying it to Christ and the Church."240 And the Lord himself says in the Gospel: "So they are no longer two, but one flesh."241 They are, in fact, two different persons, yet they are one in the conjugal union, . . . as head, he calls himself the bridegroom, as body, he calls himself "bride."242
230 St. Augustine, In Jo. ev. 21,8:PL 35,1568.
231 Pope St. Gregory the Great, Moralia in Job, præf.,14:PL 75,525A.
232 St. Thomas Aquinas, STh III,48,2.
233 Acts of the Trial of Joan of Arc.
234 Jn 3:29.
235 Mk 2:19.
236 Cf. Mt 22:1-14; 25:1-13; 1 Cor 6:15-17; 2 Cor 11:2.
237 Cf. Rev 22:17; Eph 1:4; 5:27.
238 Eph 5:25-26.
239 Cf. Eph 5:29.
240 Eph 5:31-32.
241 Mt 19:6.
242 St. Augustine, En. in Ps. 74:4:PL 36,948-949.
243 St. Augustine, Sermo 267,4:PL 38,1231D.