The Firefox version that has the "managed by your organization" is a package build from your Linux distro. The updates are provided in the package manager by the said distro and are not from Mozilla. Mozilla only provides updates to their own builds of Firefox.

The firefox-116.0.3.tar.bz2 you downloaded is just a archive as you untar it and then run the firefox-bin in extracted firefox folder to start Firefox. There is no compiling or a installer file.

You need to make sure the Firefox folder has read/write permissions for the user to get Firefox updates from Mozilla so a easy way is to have the Firefox folder in HOME. You can then make a lamcher shortcut to desktop or panel or menu etc to the firefox-bin

https://support.mozilla.org/en-US/kb/install-firefox-linux

P.S. In the config data you shared, it shows you are neither on the ESR channel nor the regular release channel, but a Mint-specific one:

"updateChannel": "release-cck-mint"

See also:

• https://blog.linuxmint.com/?p=4259

See also:

• https://blog.linuxmint.com/?p=4259

Firefox cannot handle its own updates.

When it can handle its own updates if you are using the official builds of Firefox for Linux from Mozilla ;)

Take your pick. https://www.mozilla.org/en-US/firefox/all/#product-desktop-release