Symbolism, Imagery, Allegory
It's pretty clear that these little artifacts are significant when the Officials decided to march around one night and take everyone's away. In a Society where every single thing in your life is allocated and given to you by the government—and everything you own, eat, and wear is identical to everyone else's things, food, and clothing—having something of your own is pretty important.
Cassia's got a compact, Bram has a watch, Xander has silver cuff links, and Ky has a compass. And these are the only things in their lives that are theirs, that make them unique from others and that can represent who they are as individuals. Beyond that, artifacts are passed down through families, so they're also people's sole links to their ancestors, and to history—which has largely been wiped away and rewritten by Society.
They're such an important piece of people's identities, in fact, that the government decides that, "artifacts promote inequality among members of Society" (18.57) and they confiscate them all. When they do, we understand that they are taking away people's individuality and connections to their familial past. It's a crack in the rosy façade of Society.