A burger-based currency would possibly be more stable than some of the things actually underpinning the global financial system. However, the Bank of Burgers doesn't actually provide one, seeing as it is not an actual bank, just a trendy Tackleford restaurant that is part of the artisanal hamburger craze that has gripped that city. Amy's long-time assistant Melanie quit to work there as a waitress. In an alternate timeline seen in "The Case of the Forked Road", Amy worked there too after giving up on running her own store. Back in the main timeline, though, Amy stuck to storekeeping and hired Shauna as a part-time assistant after Melanie's departure.
In the alternate timeline, Shauna and Mildred ate there in order to give Melanie a flyer for an appearance by Ryan's band Chowder, in the hopes that she'd bring Amy and that this would lead to Amy and Ryan getting together romantically (they were separate in this alternate world, unlike the main reality where they are married). As Mildred is a vegan, she ordered a meatless burger (the Dimbleby), while Shauna ordered an Eddie Mair. (The burgers appear to be named after British broadcast journalists.) Shauna ate there again later when she was in a state of frustration after the other girls temporarily gave up on working with her to solve the time-travel-induced problems with the timeline.
More recently, and once again in the "normal" timeline of reality, this was the meeting place arranged by Lottie and Mildred for Shauna to meet up with her estranged father, something Shauna was not consulted about in advance. There, Shauna learned her "origin story", though she seemed more interested in the burger (which she had some difficulty eating). Around the same time, it was the place Desmond Fishman suggested to Claire Little for an eating place, but she took him to Kale & Hearty instead.
In the real world, Burger King Russia is introducing a cryptocurrency, "Whoppercoin", so perhaps life is imitating art.