This seems to have worked for me: close Opera, go to C:\Users\AppData\Roaming\Opera Software\Opera Stable (got this path from the About page), open the "Preferences" file in your text editor of choice, look for "checkdefaultbrowser", change its value from true to false, save your changes, and start Opera. I've restarted Opera several times and the Settings window didn't come up again. The value in the Preferences file switched back to true (I guess the first time I opened Opera again) but the Settings window is not coming up anymore.

Thank you SO MUCH, alexvy86! This worked immediately for me in Windows 10. This was becoming very bothersome... I appreciate your sharing more than you know. Have an awesome day!