Sorry to resurrect a zombie thread but does this help? (from Opera help page):
"Setting cookie preferences gives you control of how Opera handles them. The default setting is to accept all cookies. To set your cookie preferences:
Go to Settings .
Click Advanced in the left sidebar, and click Privacy & security .
Under Privacy and security , click Site settings .
Click Cookies and site data .
At the top, turn on or off Allow sites to save and read cookie data (recommended) .
There are two additional options for handling cookies:
Clear cookies and site data when you quit Opera – Deletes cookies that have been added after this preference was enabled when closing the browser.
Block third-party cookies – Blocks cookies set by other websites whose content is displayed in frames or images on the current website you are visiting. Blocking third-party cookies will allow only the cookies for the site you are visiting to be set on your system.
Click the Add button next to the Block , Clear on exit , or Allow sections to set site-specific cookie preferences for websites.
Type [*.] before the domain name to create an exception for an entire domain name. Creating an [*.] opera.com cookie exception will apply to pages such as blogs.opera.com and addons.opera.com . IP addresses can also be added as exceptions. "
So first check Allow sites to save and read cookie data (recommended) is turned on.
Then just make sure YouTube hasn't added to the Clear On Exit section.