Is it possible stable coins could have their own bank since they are backed by a country's currency like the U.S Dollar which they use to maintain a stable value?
For any coin to have its own bank they don't need to be "backed" by anything. The criteria is
to be centralized which stablecoins already are. That means the answer is yes they can have their own bank.
After all the dollar itself is not backed by anything and it has its own banks simply because it is centralized!
This is exactly why there is a flaw in this design. You see centralization and blockchain technology don't mix well. It is like addition of contradictions!