I don't believe anyone who speaks of "unity in diversity" is arguing for carving up the body. Not at all. But the old 1950's Church of Christ doctrine that we must all think and believe and look exactly alike before we accept each other is simply not Biblical.
HCSB 1 Cor 12:13 For we were all baptized by one Spirit into one body—whether Jews or Greeks, whether slaves or free—and we were all made to drink of one Spirit. 14 So the body is not one part but many.There is only one body. But in Acts 15, the Torah-keeping Jews and the non-Torah-keeping Gentiles, who made up that one body, were different, to such a degree that if we were to see these two groups today in buildings across the street from each other, we would refer to at least one of them as a "denomination" and condemn them as sectarians, whereas the Scriptures record the Holy Spirit as seeing such an arrangement as "good" (Acts 15:28).
Should we call evil that which the Holy Spirit calls good?
I believe we do this because we still think we're supposed to look exactly alike, and believe exactly alike, and sport the exact same name. But the Scriptures show that's not the case. The body is not one part but many (1 Cor 12:14). This is true on an individual level (1 Cor 12:27), on a race level (v. 13), on a social-status level (v. 13), on an ability/giftedness level (Rom 12:3ff), on learning-styles level (1 Pet 3:1 word vs demo; Acts 21:10-11 skit vs words only), and on a different doctrines/practices level (Acts 15 & Rom 14).
The goal is not division, or branding; the goal is unity, despite division and/or branding. We don't want to carve up the body into different brands. But in order to follow the Biblical example, in converting or interacting with members of another group, we must be willing to allow that group to retain its pre-existing distinctiveness, allowing them to differ from us, significantly.
We've typically interpreted this "many parts" passage as having very limited range: you can be an elder, while I'm a deacon, while sister Sheri is the Joy Bus driver, but we still must believe the same thing about Pre-destination and the nature of Hell/Hades/Paradise/Heaven and when the Lord's Supper is to be observed and by whom. But the Scriptures say this range is broader than that: you can meticulously observe the Torah of Moses (Acts 15, esp v. 5), observing holy days and dietary restrictions such as avoiding ocean scavengers as food, while I could care less about some day of the week being more special than another and consider everything as clean, including shrimp and wine (Rom 14).
Just look at those particular differences: what if you belonged to a group that insisted on meeting every Sunday morning, and total avoidance of alcohol, and total avoidance of pork products, and your group met in the old Safeway store on 3rd street, and you had a sign out front that said "church of Christ", whereas I belonged to a group that met every Saturday night after sundown (believing the Jewish time-keeping system is more Biblical), and we had a meal together that was often composed of BLT sandwiches, and that after midnight (as per Paul's example in Acts 20, which, btw, puts his Lord's Supper observance on Monday for you Gentile time-keepers), we had some unleavened bread and alcoholic wine in remembrance of the Lord's death, and we met in the old Baptist assembly building on 4th street, having painted over the word "baptist" and left the sign just saying "4th Street ...... church"? To many in the brotherhood, one of our groups (probably mine) would be a "denomination". But as long as your group and mine got along as part of a united one-body, it seems that the Holy Spirit would be pleased that you don't force your scruples on me and I don't force mine on you and that despite our differences, we are brethren.
What matters is not whether we have the same function (1 Cor 12:27ff), or have the same doctrines on unclear issues (Rom 14), or look/dress the same (James 2:1ff), or even agree on what we're going to call ourselves when we gather together ("the Way", "the church of Christ", the "church of God", the "Circumcision", etc); what matters is "that the members would have the same concern for each other" (1 Cor 12:25), so that "there would be no division in the body" (v. 25).
In other words, Unity is not based on having the same "brand", or even the same practices/thinking, but on getting along as a unified group; this is the greatest, "better way" -- Love (1 Cor 12:31--> chapter 13), regardless of our differences.