but if all you want in some site's icon, you can always try to download it with okhttputils2 from, eg, https://www.b4x.com/favicon.ico.
if the icon is kept somewhere else, you'll have to track it down first.
the web share api does not work in webview. there is no question of "enabling" it or "making it work" or opening a "sharing page". it is not supported by webview.
if you want to share resources with other apps, that is a different matter. you can download a web icon and share it with another app.