After stumbling around for a while trying to fix this error, I changed permission in one of the parent directories and finally got an error message in Opera that mentioned that I didn't have permission to change the profile. And Opera wouldn't open. Now I was getting somewhere! Something to actually fix!
Changing the permissions of the folder mentioned in the error message fixed this issue. The error message no longer appears when I open Opera.
Unfortunately, I got interrupted around that time and I've forgotten which folder it was. But I never did find any mysterious file or folder that could be called 'Profile'. I guess it's what Leocg says, the entire directory is my profile.
Other than the permissions for a folder that I hadn't changed prior to the error message needing to be changed, I can't think of anything unusual that might have caused this error.
SOLVED