The Google accounts are not required for e-mail.

The banks have deactivated the hardware devices that provided one-time access tokens and they now require for login an app installed on your smartphone, and they refuse to give you the app from any other source except the Google store (or Apple store). You cannot download any app from the Google store without a Google account.

It would be enough for the banks to provide the apk file for downloading on their Web site, to be able to avoid having a Google account.