Installing programs in git bash?

You can use chocolatey or scoop which will allow you to install software from the command line and they do work on git bash. But if you really want linux software you can install windows 10 then the windows linux subsystem which does allow you to use apt-get if you choose Ubuntu over Opensuse.

Git Bash does not seem to have a package manager.

When using Cygwin or MobaXterm (and there may be more derivatives), you actually can use apt-get like it's ubuntu.