This is worth stressing: such regulations often exist to prevent fraud from outright scammers. If such a regulation did not exist, I am confident there would be institutions that called themselves a "bank" but met none of the basic consumer protections we expect from a bank.
Generally, it's a good idea to force labeling of businesses and products to match their potential customers' expectations of what they mean. It's one of those places where fostering a well-functioning free market (by decreasing information asymmetry, to put it succinctly, even if we're not talking outright scammers) is at odds with some strict notions of liberty (to call your business or product whatever you please).
There are similar requirements in some jurisdictions for lawyers. For instance if you work for a law firm, but are NOT a lawyer, you’re business card must indicate that.
Given the crisis of 2008, I would say that this failed utterly, wouldn't you? It's just another few millions lost in regulation compliance to add to the billions lost to the "outright scammers" that are the regulated banks.
