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.
