Currently, there is only a single Private Browsing session available. All Private Browsing tabs and windows share the same set of cookies etc.
For web programming E2E testing it would be very beneficial to be able to have multiple Private Browsing sessions in parallel.
Proposal for a simple implementation solution:
All tabs of a Private Browsing window should share the same cookie/website data context; whereas a new Private Browsing window should create a new, blank cookie/website data context.
So, given you have two Private Browsing windows, two sets of cookies/website data would be available, and all of a Private Browsing window's tabs would share that window's context.