You must be thinking of Windows. Unix did it right, and then later, Windows came along and developed wrong ways of doing things.
With Windows, replacing a file that is in use by a running process can badly affect that process. The process will reference locations within that file and get incorrect information from it, usually with catastrophic results. That's why a Windows update generally requires a reboot to ensure that all processes are using correct versions of libraries etc.
With Unix, once a file has been opened by a process, that same file will always be available to the process even if the original file is removed from the filesystem.
After an update, the filesystem will contain a different version of the file, and all process that start after the update will use that new file. But, unlike Windows, all old Unix processes will continue using the original files that they started with. Even though no longer accessible via the filesystem, those files will persist as long as any process is using them. Eventually, when no processes are using the files, the old version of the files will finally be deleted.
You may of course decide to restart Firefox (or other processes) if you want to get the benefits of the update right away. The choice is yours.