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?
Overall it's always possible. As we all know stablecoins have a lot of issues, especially decentralization and regulations.
So the major problem here is the regulation. As you can see the current famous stablecoins, Tether(USDT) and Circle (USDC) are already like banks, and they already encountered a lot of issues recently.