I have been trying to avoid BankID and I got me a “Skatteverkets id-kort” for this purpose, it has an electronic id built into it, “e-legitimation från AB Svenska Pass”: https://www.skatteverket.se/privat/folkbokforing/idkort/elegitimationpaidkortet.4.3810a01c150939e893f8169.html – the “AB Svenska Pass” company is in practice part of Gemalto. That e-id works in Linux, so I can login to Skatteverket in Linux, without need for BankID. Then I have been looking for a bank that supports this kind of login, but not found any such bank. So far I can only login to Skatteverket using that e-id but in principle other actors like banks could support it if they wanted to, I think.
Then there is “Freja eID” but it seems they don’t support Linux.