I have Opera on several devices, and I have sync enabled and it is set to sync everything, and on top of that I manually went trough all the settings, and all the options are identical, yet on one of the devices it logs me out of everything every single time I quit Opera. It logs me out of Opera too.
And I did check, I do not have Clear cookies on exit enabled.
This is clearly a bug, and I hope it will get fixed, because forums are full of complains of this particular issue.