https://support.mozilla.org/en-US/kb/private-browsing-use-firefox-without-history#w_what-does-private-browsing-not-save

Firefox can use the Password Manager to enter passwords when you are in PB mode, but you can't store new passwords.

A detail like a website remembering you (log you in automatically) is stored in a cookie. It is not possible to make a website remember you across a session because in PB mode all cookies are session cookies and expire automatically when you close all private browsing mode windows.