I understand where you're coming from, but this is a usability nightmare. Which cookies will other tabs in the same window have? If you open another instance of email, which account will be signed in? How do you indicate which profile is in each tab?
Separating profiles into windows drastically simplifies the situation.
There are two groups. Regular and private. All regular ones share session, so do all private. I never had a problem with that setup. Though I wouldn't mind if private tabs would each had its own session unless they were opened from following a ling from other private tab. But that would be definitely harder to indicate to the user.
The issue isn't defining behavior - it's communicating that behavior to the user. It's much easier to prevent confusion ("wait, I did what in a regular tab??") when private tabs are in separate, very differently styled windows.
Separating profiles into windows drastically simplifies the situation.