If I am logged in to a site in a regular tab and open another regular tab and go to that site it sees I am already logged in and goes to my account.
So then if I open a private tab and go to that site it does not see I am already logged in so I have to log in again to use a private tab.
But now if I then open a second private tab and go to the same site it now sees I am already logged in.
This has to mean although private tabs are isolated from regular tabs, private tabs are not isolated from each other.