Then maybe take a trip to the local branch of your bank (if you have one) and speak customer services or a member of staff there...

My bank provides card readers that create a code that are entered at these prompts as well as logging into the bank site itself. I think the only alternative is to use a phone. It may be the case that none of the banks in your country provide things like card readers but maybe ask around. That is if switching bank accounts or creating another bank account for online purchases is something that you are happy to do.

I'm pretty sure these changes are part of the EU requiring it something like 3d secure 2.0 https://en.wikipedia.org/wiki/Strong_customer_authentication
Basically its a this will make things safer, put together by those eager to make money and ensure that they have no liability when their shoddy design/implementation fails and proves to do the opposite of what its supposed to.

Sounds really bad, I hope things get sorted out soon and If possible: Please share the final answer

Probably you already did it, but I'd suggest: Try from other device, different internet connection, different browsers.
Loose some config on your ad tracker and script blockers, etc

Given that the page you've linked to says "designed in such a way as to protect the confidentiality of the authentication data", I'd suggest a formal complaint to the payment processor - or directly to the regulator if the payment processor doesn't have an obvious way to complain to them.

Wikipedia's link to the source doc is broken, it should be https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=CELEX%3A32015L2366