hello, the request for a master password should kick in once you are on a password form that should auto-complete with your username and password or once you try to access the stored passwords through the firefox options > security panel.

Use a Primary Password to protect stored logins and passwords

when a website uses a cookie to recognise you in a new browsing session and logs you in automatically, this is not an intended use-case for the master password. in this case you'd have to alter the cookie settings in firefox or specifically for pages where you don't want to get automatically signed in: Enhanced Tracking Protection in Firefox for desktop

The Master Password is only used to encrypt names and passwords that are stored in the Password Manager and doesn't provide additional protection.

Details like websites remembering you (log you in automatically) are stored in a cookie.
If you reopen pages automatically via session restore (Show my windows and tabs from last time) then the cookies are stored in the sessionstore.js file as part of stored session data.

See http://kb.mozillazine.org/browser.sessionstore.privacy_level