Do you use Opera Sync?

Are the time, date, time zone and daylight savings time settings correct on your computer?

Goto the URL opera://settings/siteData, search for youtube.com and choose "remove all shown". Do the same for google.com. Then, goto the URL opera://settings/passwords and clear your google.com password if it's there. Then, restart Opera and try logging in again and choose to save your password after a successful login if you want and it give you the chance.

Goto the URL opera://extensions. Do you have any extensions installed? If so, temporarily disable them to see if you can find the culprit.

Try in a private windows in Opera to see if there's any difference.