According to Catholicism, is the Body of Christ the same as the Bride of Christ?

On one hand, they refer to the same entity, the church. On the other hand, when the Bible uses these terms, it refers to different aspects of the church. In brief, the Body and the Bride both refer to the church. However, the Body emphasizes how the church fully expresses Christ as the church, and the Bride emphasizes how the church is Christ's intimate, loving match and counterpart.

To understand these differences in a larger context, it helps to see the seven distinct designations of the church in the book of Ephesians. The entire book is needed to understand these points in context, but I will cite only a few key verses that highlight them most strongly. In response to your question, please particularly note the points on Body and Wife:

1. Church: Eph 1:22 And He put all things under His feet, and gave Him to be head over all things to the church,

   "Church" translates the Greek ekklesia, meaning called out. This word refers to the assembly of those whom God has called out of the world to accomplish His own purpose, as detailed in Ephesians 1.

2. Body: Eph 1:22-23 And He put all things under His feet, and gave Him to be head over all things to the church, which is His body, the fullness of Him who fills all in all.

   Christ is Head whom God has established over all His creation. The church is the Body of this head, the fullness of Him who fills all in all. Just as our physical bodies are the full expression of the personality, will and actions of our physical heads, the Body of Christ, when functioning properly, is the full expression of the personality, the will, the desire, and the actions of Christ the Head.

3. Citizens: Eph 2:19 Now, therefore, you are no longer strangers and foreigners, but fellow citizens with the saints and members of the household of God,

   We are citizens of God's kingdom, with the rights, protections and responsibilities there pertaining. We have the rights and enjoyment and blessings of the kingdom of God, and the obligation to be faithful to our King.

4. Family members: Eph 2:19 Now, therefore, you are no longer strangers and foreigners, but fellow citizens with the saints and members of the household of God,

   We are God's own children, born of His life and sharing His divine nature. We are the members of His family, brothers and sisters to each other, equal without distinction.

5. Building of God: Eph 2:21-22 in whom the whole building, being fitted together, grows into a holy temple in the Lord, in whom you also are being built together for a dwelling place of God in the Spirit.

   We are God's house, His building, His real temple, His dwelling place in S/spirit. We are being built up together as one, not divided and scattered, to become God's home where He can rest for His satisfaction.

6. Wife of Christ: Eph 5:31-32 “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.

   As Eph 5:21-33 expound, we are the bride of Christ, His wife. Just as Eve was created out of Adam in his sleep, from his flesh and bone, we are created out of the death of Christ with His indestructible resurrection life (bone) in His resurrected human nature (flesh). We are His love, His counterpart, His match, His completion. Without us, He is only an incomplete half. Only in perfect intimate union with us can God say, "My new creation is very good."

7. Warrior: Eph 6:10-11 Finally, my brethren, be strong in the Lord and in the power of His might. Put on the whole armor of God, that you may be able to stand against the wiles of the devil.

   We fight the spiritual warfare against God's enemy, not by weapons of flesh and not individually, but corporately as one Body fighting through prayers in the S/spirit.

Concerning the second part of the question:

Eve was taken from the Body of the First Adam, will the Bride of Christ be taken from the body (Church) of the Second Adam (Jesus Christ)?

Concerning "Second Adam", I am not aware of such an exact expression in the Bible. The closest I can find is 1Co 15:45-49 where Paul calls Christ the "last Adam" and the "second man"; these expressions are related, but they refer to distinct aspects to Christ. In brief, Christ as the last Adam is the one who terminates the old creation; Christ as the second man is the one who begins a new race of men, the resurrected men in the new creation. Let me quote from the footnotes of the Recovery Version of the Bible, 1Co 15:47 note 2:

Christ is not only the last Adam (v. 45) but also the second man. The first Adam (v. 45) is the beginning of mankind; the last Adam is the ending. As the first man, Adam is the head of the old creation, representing it in creation. As the second man, Christ is the Head of the new creation, representing it in resurrection. In the entire universe there are only two men: the first man, Adam, including all his descendants, and the second man, Christ, comprising all His believers. We believers were included in the first man by birth and became a part of the second man by regeneration. Our believing has transferred us out of the first man into the second. In regard to our being part of the first man, our origin is the earth and our nature is earthy. In regard to our being part of the second man, our origin is God and our nature is heavenly.

I believe that the correct way to interpret the Bible is to use the Bible to interpret the Bible (hence my Biblical references from Ephesians for the points above). Neither in this passage on 1Co 15:45-49, nor in the larger context of 1Co 15, do I see any reference to Eve or to the church being a match or counterpart or wife of Christ. Paul's focus in 1Co 15 is that Christ, the second man, is the prototype of human nature in resurrection (that is, the new creation). His emphasis is that individual believers in Christ (as distinct from the church, Body or Bride as a corporate entity) will physically become the same as the resurrected man Jesus, having bodies that are physical yet spiritual, not soulish as with our present nature from the first man Adam. Coming back to your question:

Eve was taken from the Body of the First Adam, will the Bride of Christ be taken from the body (Church) of the Second Adam (Jesus Christ)?

It seems that you are asking if one can separate the Bride of Christ from the Body of Christ. I don't see any biblical basis for such a separation. When the Bible talks about the Body of Christ (e.g. Eph 1, Rom 12, 1Co 12), it does not simultaneously talk about the Bride. When the Bible talks about the Bride or Wife of Christ (e.g. Eph 5, Rev 19, Rev 21, Rev 22), it does not simultaneously talk about the Body of Christ. These are different aspects of the same church, so while they are both valid, true and real, they are not separate such that the Bride can be taken out of the Body as a separate entity.