I think it isn't possible.
The closest thing to this is the command line switch "--user-data-dir" but that will share the whole profile (cookies, history, session, bookmarks, etc).
E.g. this is in the Opera shortcuts, should be set to:
"C:\Program Files (x86)\Opera\launcher.exe" --user-data-dir=\\server\path\to\profile
I don't even know how you get this to work before if hypothetically you ran 2 different Operas handling the same file at the same time. I believe it'd give instant errors.