Being logged in and remembered is a function of your browser 'cookies'.
Each different browser has it's own set of cookies by design and are/should-be sand-boxed from each other by the computer's operating system.
Depending upon your checking the box 'Remember me' as said above in combination with your settings in each browser's preferences on how to handle cookies will usually determine how long your active login is remembered by the browser.
In Firefox, I use an add-on called 'Cookiekeeper' that allows finer grained control than the cookie manager built-in to Firefox, and lets me clear individual or all cookies, as well as 'protect' cookies so they do not get deleted.
Cookiekeeper, in combination with another add-on called 'Self-Destructing Cookies', which auto-deletes cookies after you close a web site lets me help to control being tracked across various web sites, along with add-ons uBlock Origin, EFF's Privacy Badger, and NoScript.
But be warned that you have to TRAIN NoScript which takes lots of time and with the above combinations, a handful of web sites that rely extensively on Javascript simply do not render web content well, or at all, and for that I use another browser if I really want to see that page (Midori, on Linux, which is similar to Apple's Safari in that it uses the Webkit rendering engine).
I have Midori set to purge all private data when closing so the cookies get wiped when I close it.
I also have other protections in place at the router (running pfsense, with pfblockerNG) and also LAN-wide at home with Pi-Hole, Squid and Privoxy, but the details of using these are beyond the scope of the OP's question, so I'll spare the tech talk for another time.