Symbolism, Imagery, Allegory
Back in Dickens' time, you couldn't just walk up to someone's house and expect them to greet you with a friendly, "Howdy-doo, neighbor?" and give you a cup of sugar. Okay, to be honest we don't really do that today… but at least we aren't hung-up on the idea of calling cards.
The whole social world in 19th-century England was card-based: you left your card with people to let them know your social status. People loved cards almost as much as the guys in American Psycho.
Dickens makes fun of this "culture of cards" at one point, writing,
All the world and his wife and daughter leave cards. Sometimes the world's wife has so many daughters, that her card reads rather like a Miscellaneous Lot at an Auction. (5.17.2)
Dickens uses the symbol of the cards to show just how much people in British society reduce one another to pure social status. They won't even look one another in the eye if their business cards aren't good enough. Talk about judging a book by its cover, right?
Even a character on the moral up-and-up, like Jenny Wren, knows the importance of having a card:
"Stop a bit […] I'll give the lady my card." She produced it from her pocket with an air, after struggling with the gigantic door-key which had got upon the top of it and kept it down. (11.2.43)
What better symbol to portray superficiality than the (almost) literally two-dimensional card? What better was to show the flimsiness of society than to have everyone's personalities boiled down to something that can be ripped in half? That Dickens, he was a symbolically savvy fellow.