What I find illogical about Bottle Caps in Fallout 3, is that they should be in Fallout 3. Bottle Caps were used in the Core Region in Fallout and the caps had value because the Water Merchants were backing it. By Fallout 2, bottle caps were replaced with coins. So why is it that the people of DC are using bottle caps? People over 3000 miles away from the core region and the people there stopped using bottle cap many decades ago. Who is backing the value of the cap in DC? No one.
Bottle caps were used in the Core Region and it can be assumed from two main scenarios as follows.
1. The earliest people who used caps in the Core Region simply walked to D.C like an expedition with a bunch of other people in search of treasures or simply may have found a terminal that has details of the National Treasury or something. Or at the very least, adventurers like the Lone Wanderer traveled to D.C and many different adventurers coming and going introduced a currency that was easy since there's the Nuka Cola factory nearby and a bunch of stores that carry Nuka-Cola.
2. Caps simply came out due to necessity. Remember that this is all too possible. Just as Pascal's Triangle came from both Europe and Asia independent of each other with only slight differences along with the fact that a monarchy came out of nearly every civilization relatively independent of each other, caps could have the same reason why. After all, Nuka-Cola factory. Hard to counterfeit unless you have lots of metal and know how to operate the factory.
As for who's backing it. Well nobody. However, if merchants start only taking caps as an accepted currency (other than simple trading), caps gain value. All money is worthless in a sense but even if a large organization isn't backing it, if smaller organizations start putting monetary value on something, everybody else must follow suit.