That's weird. That setting shouldn't affect cookie session storage at all. Usually when this issue happens, it's because Opera is set to delete cookies when closing Opera at the URL opera://settings/cookies. Or, it's because the user has CCleaner installed and set to clear Opera's cookies and local storage.
If the startup option is really affect cookies, I'd goto the URL opera://about, take note of the "profile" path, close Opera, and delete the "Sessions" folder in the profile folder to see if that clears things up. Note that you'll lose all opened tabs if you do this.
I tested though and "Begin with a fresh start page" doesn't log me out of anything when closing/starting Opera.