Seems like another case of Apple's app-artheid system -- only Native Apps in the Apple App Store get to have an icon when using Sign In with Apple. Web Apps are second class citizens, as they are in so many other ways.

I have a similar question, but it is about how to set an icon for the app/website. And a link to the privacy policy. I see native apps with such, but mine is a web app which uses Sign In with Apple. Is there a way for me to put an icon in the "Apps using Apple ID" dialog box?

As of iOS 16.4 in March of the year 2023 the web Push API is supported. Yay! -- but also, about D.T.