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?
Pooya explained it well what you are describing.
Or also, knowing that stablecoins are different from cryptoassets which do not have assets as backings and are more volatile, will in the future, the success of stablecoins not call for something like a bank or a recognized office building built to show legalization and acceptance as a means of payments for transactions and services?
They're actually working as that right now. And there will be more and bigger institutions that will make their own stable coins. Lately, we've seen that PayPal launched their own stable coins, PYUSD. They're not a bank but they function like a bank, a storage, an exchange and as well as remittance for everybody.