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How the twins personify society is really approached in Chapter Six. Their childhoods were difficult; they had been sold into slavery to pay off the debts of their parents, and they had begged for food when they had none. Through their own devices and initiative, the twins work to but their freedom, and then go onto become respected citizens (as evidenced in the town's treatment of them in Chapter 7).
The way in which they personify society is the comparison that can be made between the twin's childhood and the experiences of America's southern slave popuation, but unlike the way their lives parallel those of the slaves, the slaves don't have a way out. Even those that manage to escape will never have the opportunity to become self made successful men or to free themselves from the oppression they experience because of the color of their skin.
Gradesaver also has this quotation right here on their website. It further explains treatment due to race and/or appearance;
Twain notes that one of the Capello twins has a darker complexion than the other. Of course, the townspeople make no distinction between the two brothers in response to this difference in skin tone. Doing so would be irrational, as the twins are otherwise "exact duplicates." Yet, inexplicably, the town does draw a distinction between a pair like Tom and Chambers based on race, even though they are so similar in appearance that Roxy is able to switch them without attracting anyone's notice.