That is weird. I also have BankID but I never got a fullscreen SMS. It just gives me a normal iOS app notification on the top, which I click on, and then the app opens.
You must be using the new BankID app which sends push notifications, but I don't think it's supported by every site which uses BankID.
Some BankID providers offer a regular bespoke ios/android app that replaces the physical dongle, which can operate in two modes, either by displaying the one-time code or by using the internet to autofill the code.