SMS authentication is... well by one reading of PSD2, it's not acceptable. But in real world, it is basically necessary, and not _that_ insecure (if you ignore SIM swapping attacks etc.). The WYSIWYS aspect comes not from the code but from the message text, which is crucial (and per PSD2, should include at least the amount and... receiver? I forgot). But sure, if people don't read or understand the message, it's not ideal...
While FIDO provides better phishing resistance (than SMS, not necessarily than authentication apps), it doesn't protect against transaction modification (e.g. man in the browser) and for people who care about and understand security, it is strictly worse.
'man in the browser' seems like a situation where the user's device is compromised. In that case it is not big stretch that not only browser could be compromised, but also SMS reading app is compromised.
I.e., the reasonable security request should not be security against 'man in the browser', but security against 'user device is compromised'. In that case SMS is worse, as attacker could completely bypass it, while for FIDO it still need to phish the user to press the button.
Very dubious. The trick to phishing is that humans are easily confused about what's going on, and WebAuthn recruits the browser to fix that completely. Your browser isn't confused, the browser knows it is talking to fakebank.example because that's the DNS name which is its business, even if this looks exactly like the Real Bank web site, perfect to the pixel and even fakes the browser chrome to have a URL bar that says realbank.example as you expected.
I don't see bank authentication apps helping here. It's very easy to accidentally reassure the poor humans everything is fine when they're being robbed, because the authentication part seemed to work.
I'm somebody who really cares about and would like to think they understand security very much, and I don't think it's strictly worse at all.
One of the things banks have an ongoing problem with is insider facilitated crime. Which means secrets are a big problem, because the bank (and thus, crooked staff working for the bank) know those secrets. Most of these PSD2 "compliant" solutions rely on secrets, and so are vulnerable to bank insiders. FIDO avoids that because it doesn't rely on secrets†.
† Technically a typical Security Key has a "secret" key [typically 256-bit AES] baked inside it, but a better word would be symmetric rather than secret, there is no other copy of that symmetric key, so it isn't functionally secret.