Recently I find that, when logging into a website, I'm not invited to touch ID

@Moorlander

I would suggest heading over to System Preferences > Touch ID: Uncheck/disable the box for Unlocking your Mac and then enable it. You will need to enter your macOS user password in the authentication prompt. Follow that with checking 1Password to make sure Touch ID is enabled within the app. 1Password > Preferences > Security. Give that a try and let us know how you fare?

Also in Safari toolbar right-click on the 1Password icon select settings. Make sure integrate with the app is enabled in the General section.

Can you confirm you're using 1Password in the browser? In your details you listed 8.9.7 however 1Password in the browser is number 2.3.8 for Safari. If you're not using 1Password in the browser, you would want to install 1Password in Safari on your Mac for use with 1Password 8. Please open it at least once and follow the directions being sure to all it access to all sites. We need that permission to read (save details), write (fill details) on your behalf.