1Password for Ungoogled Chromium

Sure thing.

The new extension works regardless of the desktop app so you can install it on any browser, even if the browser is not an officially supported one by the desktop app. Only the desktop app cares which browsers are officially supported and which aren't.

So let's take the desktop app out of the equation here.
You can install the new extension in your Ungoogled Chromium browser and it will work as you expect it to work - you can unlock it, autofill on your websites, save new logins, generate passwords etc.

The only downside of not having it integrated to the desktop app is that the extension will need to be unlocked separately every time you open your browser, and you won't be able to use biometrics to unlock it - you will have to use your Master Password to unlock it.

I hope that clarifies things better :)