You aren't running Firefox in permanent Private Browsing mode (Always use Private Browsing mode; Never Remember History) as that works differently on Mac ?

Note that on Mac closing Firefox can be more complicated because you can close all open windows, but keep Firefox running with the menu bar visible or minimized in the Doc.

Not sure whether this is relevant to your scenario, but for general reference:

Regular windows and private windows use two separate cookie jars. All of the regular windows share one (disk-based) cookie jar, and all of the private windows share one (memory-based) cookie jar. As long as any private window is open, other private windows can still use the cookies in the memory-based cookie jar. Only when the last private window is closed does Firefox destroy that cookie jar, so that when you launch the next (first) private window, you are starting from scratch.

Also, you should log out of sites if your security on those sites is important to you. Ending your session on the server by signing out protect against resumption by a hijacker who gets your cookies.