Private Browsing does not save your browsing information, such as history and cookies
There’s a lot of things private browsing does not do (partly because there’s a lot more parts of the stack involved in privacy than just your web browser) but deleting cookies on exit is literally one of the things private browsing specifically does and is directly controlled by the browser
Press X to doubt.
https://support.mozilla.org/en-US/kb/private-browsing-use-firefox-without-history
There’s a lot of things private browsing does not do (partly because there’s a lot more parts of the stack involved in privacy than just your web browser) but deleting cookies on exit is literally one of the things private browsing specifically does and is directly controlled by the browser