I think this is a problem independent of the dual screen usage but I wanted to include it in the title anyway, since finding solutions to this kind of specific problem can be hard.
When I close all of the tabs for my browser, the start page comes up, and this is something that I enjoy.
For my work setup, I use two screens, with an Opera window on each one. When I close all of the tabs on one window, I don't end up with the Start Page, since on the other screen there is still an Opera instance running. Only when I close all of the tabs on that window do I get the Start Page. I basically want to separate these two instances of Opera from each other completely, so that I can end up with the Start Page on both screens when I close all of the tabs on that specific screen.
I hope I was able to convey my problem, and I know it's very very stupid of me to even call this a problem, but if there is some way to do this I'd be so happy.