I think what you call auto-login is a combination of 'form history', which remembers the content of input fields, and 'password remembering'. Both are stored in the profile; if you get Firefox and TenFourFox to use different profiles then each will operate independantly in that respect. (Modern versions if Firefox can sync both between devices.) If you backup the profile (e.g. xxxxxxx.default) it will backup all that.

Mudcat just uses a cookie to remain signed in - a technology that hasn't changed since the 1980's. But Firefox will also remember the signon password if you want.