As is so often the case with questions such as yours, it's not a matter of "either/or" but "both/and."
Let's put it this way: the comparison between the church universal ("the holy catholic Church"--the Apostles' Creed) and a body is both an analogy and a metaphor. The expression of the church as a body is presented perhaps no more clearly than in 1 Corinthians 12:12-26, where we read,
"For by one Spirit we were all baptized into one body, whether Jews or Greeks, whether slaves or free, and we were all made to drink of one Spirit. For the body is not one member, but many. . . . Now you are Christ's body, and individually members of it. And God has appointed in the church, first apostles, second prophets, third teachers . . ." (vv.12-14, and vv.27,28a).
Notice that the analogy comes first:
". . . as the body is one . . ." (v.12).
The word as, as I was taught since elementary school is one of the words which signals a simile (the other word being the word like). For example,
"As one whom his mother comforteth , so will I comfort you; and ye shall be comforted in Jerusalem" (Isaiah 66:13 KJV).
"Abide in me, and I in you. As the branch cannot bear fruit of itself, except it abide in the vine; no more can ye, except ye abide in me" (John 15:4 KJV).
[Notice: in each of the above verses, the emphasis is mine!]
Think of an analogy as an extended simile, or a simile that is expanded to include at least one analog, or point of comparison, but usually two or more analogs, or points of comparison. The analogy of the vine in John 15, for example, indicates in "Miller's analogy-type" format,
Grapes:vine::fruit:abiding (or, "grapes are to vine, as fruit is to abiding").
[Logicians: please don't correct my Miller's analogy. I know it's not perfect!]
In a less formal way, we could go on to say that abiding in the vine and drawing sustenance from the branches of the vine via the root system is analogous to abiding in Christ and drawing our spiritual sustenance from Him. In other words, the water and nutrients from the soil which nourish the vine are analogous to Christ's power and purity in us as we attach ourselves by faith to Him.
If a poor interpreter were to force the analogy to include the idea that all Christians must paint their faces purple so that they more closely resemble grapes, then we know the analogy has been pushed too far! All analogies break down at some point, but if not pushed too far, they serve as an excellent pedagogical too.
A metaphor, on the other hand, is perhaps both a little easier and harder to understand than an analogy/simile. First of all, a metaphor does not clue us in with the word "like" or the word "as." In 1 Corinthians 12, verse 27, Paul says,
"Now you are Christ's body . . .."
We know, of course, that neither the worldwide church of Jesus Christ nor any given local church is literally Christ's body. Christ, as a flesh and bone human being ascended into heaven 40 days after His resurrection. That's why "you ARE Christ's body" is a metaphor, or as we learned in English class, a comparison which does not use the word "like" or "as" to signal its presence.
Instead of, or in place of, His human body on earth, Jesus has given us His Spirit to comfort, teach, empower, counsel, lead, guide, rebuke, and encourage us.
Whereas in the verses 12-26, Paul uses an analogy, in verses 27 and 28, he uses a metaphor, and he makes it crystal clear that the body is the church; the body of Christ = the church of Christ:
"Now you are Christ's body . . . and God has appointed in the church . . .."
Are we Christians literally the body of Christ on earth? No. We do function, however, as Christ's hands, feet, eyes, ears, tongues, and more, when we perform His bidding through His Spirit (the Holy Spirit) within us.
What is the biblical metaphor for Christ? For one thing, He is the head of the body.
"And he is the head of the body, the church: who is the beginning, the firstborn from the dead; that in all things he might have the preeminence" (Colossians 1:18 KJV).
There are, of course, many other metaphors for Christ, and He gave us a number of them in His teaching in the Gospels:
"I am the bread of life"
"I am the resurrection and the life"
"I am the true vine, and my father is the vinedresser"
"I am the Good Shepherd"
"I am the way, the Truth, and the life"
"I am the door of the sheepfold"
"I am the water of life"
In conclusion, I find your linking of the body metaphor and Christ's incarnation to be quite perceptive! If we kind of "unpack" the metaphor (which is not strictly biblical, but true nevertheless), it won't be long until we realize that yes, in the church age, prior to Christ's return, the church functions as Christ's bodily presence on earth in and through each and every one of His disciples, each of whom is a part of His body, whether a foot, an ear, an eye, or a hand.