1Password for Mac Safari works seemlessly. Edge for Mac, barely. On Safari when I click cmd-\ it prompts me to approve on my Apple Watch and then immediately works. On Edge, it sends a popup that asks for my password, but when I fill it in it doesn't do anything. It turns out I have to enter my password on the little extension icon in the toolbar. And it doesn't use Apple Watch at all, nor TouchID. I don't know what the behavior is supposed to be, but it does not behave in any way I expect.
It sounds like you are using 1Password in the browser and not our classic extension. The newer extension formerly known as 1Password X does not integrate with 1Password directly. It is capable of functioning independently. It doesn't communicate with the Mac app at this time. That's why you are unable to use Touch ID. We hope to bring Touch ID back to it in the future. One of our founders posted about it in the community. I have included the link below.
The extension also uses a different shortcut. You can also read more about 1Password in the browser.
Temporary removal of desktop app integration.