I'm having this issue. I've only recently downloaded Opera Mini (iOS 9.1). There isn't a "Report a Problem" are in the settings, but I did leave feedback.
I am logged out constantly. Closing Opera logs me out. Clicking a link on the same site logs me out. I have cookies enabled, I click the "remember me" boxes next to the login area, I save the passwords. Regardless of these efforts, nothing stays logged in and the information saving is useless.