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Gnome System Monitor is listed twice in Applications Overview. Using the Software center I determined one of them is a snap version, while the other isn't a snap. They appear to be identical in the applications Overview. Gnome Calculator is another example, and I'm sure there are more.

Can I remove the snap version and use the other one without any issues?

As the question clearly states: Why are two versions of software with identical names installed on Ubuntu 18.04? The suggested duplicate of this question does not answer the question.

By the way, the non-snap (apt version) of Gnome System Monitor is listed in the software Center twice, adding to the confusion, both listed as version 3.28.2-0Ubuntu1. Why is this happening?

Ubuntu release 18.04 new installation (not an upgrade).

Replacing some GNOME apps with Snaps is developers' decision to support modern glossy non-geek GNOME "functionality".

How to remove such snaps?
Search them by snap list:

$ snap list

Name                  Version  Rev   Tracking  Developer  Notes

core                  16-2.33  4830  stable    canonical  core

gnome-3-26-1604       3.26.0   64    stable/…  canonical  -

gnome-calculator      3.28.1   178   stable/…  canonical  -

gnome-characters      3.28.2   101   stable/…  canonical  -

gnome-logs            3.28.2   37    stable/…  canonical  -

gnome-system-monitor  3.26.0   45    stable/…  canonical  -

and then remove them with for example

snap remove gnome-system-monitor

And then install regular package with for example

sudo apt install gnome-system-monitor

Some deb-packages could be installed into the system as dependencies of other packages . For example gnome-system-monitor is a dependency of the following packages:

$ apt-cache rdepends gnome-system-monitor

gnome-system-monitor

Reverse Depends:

  gnome-core

  vanilla-gnome-desktop

  ubuntu-unity-desktop

  ubuntu-budgie-desktop

  gnome-applets

 |compiz-gnome

  cinnamon-desktop-environment



$ aptitude why gnome-system-monitor 

i   xinit        Recommends gnome-terminal | xterm | x-session-manager | x-windo

                            w-manager | x-terminal-emulator                     

p   compiz       Provides   x-window-manager                                    

p   compiz       Depends    compiz-gnome                                        

p   compiz-gnome Recommends gnome-system-monitor | mate-system-monitor         

see corresponding man-pages for the syntax of used commands (man apt-cache and man aptitude).

That's simple, one of the apps is of the tradition package manager(.deb extension) and another is of the snappy package. The tradition apps are faster in the opening, the snappy apps are slow but they provide users better control over permissions. Snap is a new format thus it still has some issues like some snap app doesn't follow your theme, slow loading, etc but at the same time, it is better than apt in providing updates.
If I create a deb package for Ubuntu 18.04 then that package will work only for 18.04 not even for 18.10. I also have to make a deb package for all versions of Ubuntu.
Snaps are not ready to replace deb packages and many people don't even want to use snap packages at all so there are both versions in store traditional as well as the snap packages.
This is what Vlc says for having 2 versions of the app on the store.

VLC for Ubuntu and many other Linux distributions is packaged using snapcraft. This allows us to distribute latest and greatest VLC versions directly to end users, with security and critical bug fixes, full codec and optical media support.
If you wish to install the traditional deb package, it is available as usual via APT, with all security and critical bug fixes. However, there will be no major VLC version updates until the next Ubuntu release.