Laws can be changed.
An entity like Amazon can lobby for a change.
Donate money to politicians likely to vote for a change.
Call for a change on the editorial page of the Washington Post.
Remember, once upon a time, there was but one phone company. Then the law changed...
Interstate/branch banking was illegal. Then the law changed.
Amazon could probably set up a members-(i.e., customers) only credit union without too much difficulty.
That was deregulated by the Gramm–Leach–Bliley Act in 1999 and likely led to the subprime mortgage crisis. It permitted the creation of the "too big to fail" superbanks.
I was able to find the law I was thinking of, and it was passed in 1956, so not terribly recent, no.
Apparently it's because the BHCA defines banks as entities marketing both checking accounts and commercial loans, and Discover Bank makes no commercial loans and is thus not, under the terms of that law, technically an actual bank.
(Also, apparently the BHCA restricts banks from operating across state lines, too.)
IIRC, there's a law expressly prohibiting any company from both acting as a bank and a retail business.
edit: Yep, there is. The Bank Holding Company Act prevents companies in commerce from engaging in banking activities.