Apparently, it becomes like this when you select decline the first time the weather's dialog is shown (where you would otherwise set the location manually or select accept and let your location being handled automatically). After doing so there was somehow no way to reconfigure it.
To resolve this limbo-like status, close down Opera via ⌘Q, open ~/Library/Application Support/com.operasoftware.Opera/Preferences in a text editor and search for
weather, which leads you to:
onboarding's value to
true and reopen Opera, which should make the weather's dialog appear again upon clicking the widget, so you can then either choose accept or enter a location manually.
On Windows, it should be analogous and the path is probably something like %APPDATA%\Opera Software\Opera\Preferences.