I had the same white, empty Startmenu square icon for Opera on my Win10 32-bit machine.
When installing Opera I had not directly accepted the the default options, but used the 'options' button in the installer window instead. For some reason now Opera was not installed in the standard C:\Program Files\Opera directory, but somewhere else (C:\Users......., you can check the install directory in Opera in Menu>about Opera)).
The following has resolved the problem for me:
- unpin the empty icon from the startmenu,
- uninstall Opera (but keep the browser settings)
- reinstall, now manually indicating to install in C:\Program Files\Opera (you may have to create this directory using admin rights)
- add (pin) the correct Opera icon to the startmenu
Please let me know if this works for you too.