For me the "Remember me" stays as long as you don't close the browser but once you close the browser you are logged out of anything from the browser. For Chrome it's "Remember Login" is different as it "AutoLogin" when you go to the saved login site. This only happens with Chrome from my uses and Firefox will logout when you close the browser and you will have to choose the correct login and pswd to login to the site again.

After you sign in to a site, it sets a cookie. Whenever Firefox requests a page from the site, it sends that cookie. This is how the site knows you already signed it. It sounds as though the cookies are not surviving after you quit Firefox. That could be for a bunch of different reasons, including:

(1) You used a private window

These cookies are not stored to disk. You can open individual private windows, or you can set Firefox to "Never Remember History" and that will make all your windows private. This feature is not compatible with staying logged in.

(2) Your Firefox is set to clear cookies when it closes

This feature is not compatible with staying logged in, unless you create site exceptions for sites whose cookies Firefox isn't meant to delete when it closes. On the Settings page, Privacy & Security panel, Cookies and Site Data section, look for the Manage Exceptions button to set up exceptions for testing.

(3) One of your add-ons clears cookies

Most users do not have this kind of add-on, but maybe it sounds familiar to you?

(4) External cleaning software clears cookies

Programs like CCleaner and Advanced SystemCare my remove Firefox cookies after they detect that you've closed the browser.

There might be some other reasons, too, but i think these would be the most common.

For me the "Remember me" stays as long as you don't close the browser but once you close the browser you are logged out of anything from the browser.

Your own Firefox is likely set to "Delete cookies and site data when Firefox is closed" in Settings > Privacy&Security > Cookies and Site Data > [v] "Delete cookies and site data when Firefox is closed".