Just kill the process or restart the computer or just logout and see if it updates.

You appear to be using a Ubuntu supplied build of Firefox 113.0 and that version leads me to think your version of Ubuntu is EOL in package repository updates. To get Firefox updates from Ubuntu you need to use a currently supported version of Ubuntu to get package updates as to why you are stuck on that old Firefox version. You can also use the Linux Firefox tarballs from Mozilla as they get internal updates from Mozilla like the macOS and Windows versions does. Just need to make sure the Firefox folder has read/write permissions for the user to get internal Firefox updates. Note if you have 32-bit Linux then Firefox has dropped 32-bit Linux support starting Firefox 145.0 and on so you should use the Firefox 140 ESR instead. https://support.mozilla.org/en-US/kb/install-firefox-linux https://www.firefox.com/download/all/