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?
I assume that we are already into it. Yeah, unlike traditional banking, these days I am able to open a bank checking account online without visiting any branches; this is very much similar to registering an account in a crypto exchange with KYC. I agree there are few differences like debitcard, check facility, overdrafting etc.. but we can find other third party services to work with cryptocurrencies to avail exact similar functionalities.
I mean bitcoin was developed to provide banking facility to each and every human without trusting any central authorities including governments and here stablecoins providing additional layer of simplicity to take banking facility to needed people.
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?
Definitely will call. But, I believe we do not need any regulation/legalization for that to make it more common among all people; only awareness is enough. But, we need to remember stablecoins are not decentralized which means they may collapse at any time or government may seize.
So, holding a big money in stablecoin is highly NOT recommended.