Same problem, with other browsers like Chrome for example favicon is displayed correctly, but not with opera. If you save that pages as bookmarks or home shortcuts no icon displayed. As i can see this happens most frequently with servers on the same lan.

@cmp-nbq What I think!,This issue is not caused by your phone but by the way Opera integrates with Android’s “Direct Share” feature, which shows the last used app next to the share button. Since you mentioned reinstalling the browser, the shortcut data likely got reset and Opera is no longer able to display the last app icon. The solution that works in most cases is to clear Opera’s cache (not full data), update it to the latest version, and make sure Direct Share is enabled on your phone. To do this, go to Settings > Apps > Opera > Storage and tap Clear Cache, then update Opera from the Play Store. After that, check in Settings > System > Developer Options that Direct Share is turned on. Finally, restart your phone. This fully restores the “last used app” shortcut next to the Share option, and the icon will start appearing again.

Проблема решена.

Нужно было зайти в свойства программы (значок) и выбрать пункт - "совместимость".
В поле "Режим совместимости" выберите пункт "Запустить программу в режиме совместимости с:" и выберите "Windows 7" или "Windows 8".
После "применить" и "ок".
Вы перезагружаетесь, и все работает.

Также, после перезагрузки, Вы можете убрать в свойствах галку с "Запустить программу в режиме совместимости с:", тем самым запуская ее от своей Windows, а не через другие версии (так браузер должен меньше лагать)

I absolutely agree