Each distro provides their methods of downloading and installing software. Mint is based on Ubuntu, and Ubuntu is based on Debian.... so you are using the "Debian family" of package management... which is why you encounter .deb files that can install programs.Another thing about the Synaptic Package manager and Software Manager can you have more than one of Synaptic Package manager and Software Manager or is one all that is needed? Are there different types?
Red Hat and others use .rpm files (RedHat Package Management).
Each distro may (or may not) use Synaptic, or their own version of Software Manager. But they will use SOMETHING so that you can add software to your Linux.
All distros can compile from source code and can install from the terminal, but the terminal commands are often different between the different distros.