Sure, just an example ING (part of ING Group, 34th bank in the world according to https://en.wikipedia.org/wiki/List_of_largest_banks only highlighting this to show it's not a tiny random "weird" corner shop) requires to use either their mobile application or ItsMe (details https://fr.wikipedia.org/wiki/Itsme if you want but basically also 2FA as a mobile app) in order to login to their consumer/professional website. You can from the mobile app scan a QR-code which in turn will ask for authentification, e.g. biometrics.

Yes indeed registering the mobile application itself requires first another form of authentication, typically an SMS confirming the number plus a physical card with a physical card reader. You then input the resulting token in the app which validates it and then you don't need the card reader anymore while you rely on the mobile app. AFAICT the physical card reader options is not offered on some mobile payment options. I do not know if they are phasing it out of if it is because another method exists, namely if you have NOT registered their mobile application as a 2FA method, can you still use the physical card and card reader. I do not know that.

To be clear they do NOT require an app per se. They do though if you want to use online services, including payments, bank transfer, reading specific kind of documents, adding specific recipients for recurring transfer, transfer above thresholds, etc.

Hope it helps. If I missed something happy to try to clarify. Also FWIW and AFAICT it's getting more and more common for online services from bank in the EU.