It's possible that other package operations are "in progress" or that your package system is in a broken state (possibly from commands you ran previously) which means hitting "Remove" in the Ubuntu Software application is simply queuing these actions and not actually executing them.
Also of note is that one of those Firefox packages I believe is a Snap package.
I have a need to reinstall Firefox without using the command line because the command line is not yielding satisfactory results.
It's worth noting that Ubuntu Software, other package managers, and most every Linux program actually are interfaces that eventually run commands on the command line. This means that if something doens't work on the command line it's not very likely to work in a GUI, except when the GUI developers have added a bit of helper code to avoid common mistakes.
You can check if Firefox is installed using:
dpkg -l firefox
If it's installed, you'll see this:
ii firefox 64.0+build3-0ubun amd64 Safe and easy web browser from Mozilla
That ii in the first column means it's installed.
It's worth mentioning that Firefox and many other applications can be added or removed, but they also store lots of files. For example, if I want I can copy my ~/.mozilla/firefox directory from one installation to another and it will keep all my plugins, cookies, etc. intact. This is because applications in Linux cannot store files in system privileged areas of the disk and therefore keep their configuration files within your home directory (~)
Separate from the files in your home directory there are commands such as apt purge which will removes a package and it's associated system configuration files but to the best of my knowledge never removes any user configuration files.
Cool, but how do I reinstall Firefox using the GUI?
I would suggest you first see if your package system is in a broken state by simply running:
sudo apt install firefox --reinstall
If this complains and says you have a broken package system then it explains why Ubuntu Software app isn't working. If it complains an existing dpkg process is already running, that also explains why the app isn't doing what you expect, and you should probably reboot Ubuntu.
Also as user pomsky mentions Firefox Quantum and Firefox are effectively the same thing now. There exists no officially supported version of Firefox that doesn't use the changes presented from Quantum, which essentially change how it uses multiple threads and other UI improvements. You can, in theory, grab a much older version of Firefox using:
apt install firefox=55.0
You probably do not want to actually do that as you'll be running an insecure version of Firefox
After removing the browser the dialog is the same: it shows a remove control. After closing the Ubuntu Software application and restarting it, it will still show a remove control.
