There appears to be a web API for this, which would be cross compatible.
If you have used Apple's service - which is completely opt-in - you could always go into an account and change the email you associated with it if they let you do that. And if you don't, this would be in the ballpark of switching email providers - yup, you'll need to change some accounts, because you've decided to no longer pay for the service from the Apple email forwarding.
I don't think that really means what you think it means. It's very similar to the way "Apple Pay has a Web API"—websites may implement it for authentication BUT it won't allow users on a Linux/Windows machine to Pay, either. This is the same thing. You'll need to use Safari on a Mac, or it won't appear.
From the developer documentation, "Sign In with Apple works natively on iOS, macOS, tvOS, and watchOS. And it works in any browser, which means you can deploy it on your website and in versions of your apps running on other platforms"
The "works in any browser" and "apps running on other platforms" parts seem to suggest it will work fine on Linux or Windows machines or even Android devices.
If I sign up for a service using Apple Sign In, is then possible to sign into my account from a Windows etc machine subsequently?