I hate to quibble, kids, but wallet chains were a staple of the leather and steel
armor affected by rebel bikers, starting in the post-WWII
era. They tended to sport tattoos, too, though body piercings didn't really catch on until much later.
For those guys, the chain was probably just a utilitarian item, but that "look" had substantial appeal for the toughs and neighborhood gang kids of the fifties. They adopted longer, heavier chains as belts, which could serve as ready weapons, in the case of unforseen rumbles.
As a stylistic accessory, though, today's wallet chains are probably more closely related to the absurdly long watch chains worn by the zoot-suiters of the thirties and forties. Far from utilitarian, those chains were symbols of estrangement from the cultural norms of society. They were badges of kinship among a rejected, and rejectionist, few.
Zoot-suit style never became a part of the general youth culture. It was always pretty much limited to black and hispanic youth who really were rejected by American society. The disaffection reached its height in 1943, with the so-called zoot-suit riots, which involved (among other things) zoot-suiters being ritually stripped of their finery, mostly by bands of sailors on shore leave.
I suppose that could happen to boy bands, but somehow I doubt that they inflame sufficient passion in the appropriate quarters. C'est la vie.