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!
Their coin or token value is backed by quality of idea from which their project initiated and their recent, current and future development ideas and plans as well as how they already fulfilled their development promises to investors through roadmaps and how will the complete their development plans in future.
Stable coins backed by gold, US. dollar or any fiat currency but except gold, all fiat currencies can be printed from thin air by governments. It does not help for a coin or token value if it is backed by something from thin air.
Gold has production cost but its value is from demands too because exchange price rate of gold has big contribution from demand, not only or mostly comes from its production cost.