If you rent a car or your residence to someone, you, as the owner, run the risk that your renter will not treat it the way you would treat your own property. By entering into the rental agreement, you, the lessor, incur moral hazard. If you doubt moral hazard exists in rental contracts, ask yourself why landlords require deposits, and why you as a used car buyer worry more about a rental car than one you buy from the sole owner.
Neither the the bank nor the depositors are behaving the way they would in the absence of deposit insurance. That is the essence of moral hazard.