Some desktop environments, such as KDE Plasma and Cinnamon, have icon-based
taskbars which group windows into a single icon.

Usually, when a pinned application is opened from this kind of taskbar, the
function of the icon switches from opening the application to maximizing
/minimizing/switching through running instances of the application. This
happens in such a way that instances of the application are grouped
together in a single icon.

It is somewhat difficult to explain, but in any case, godot3 does not seem
to behave as expected with these taskbars. Instead, once a project is opened
in godot3, the icon in the taskbar appears to duplicate, such that the running
instance of godot3 shows up separately from the pinned icon.

It seems that the fix is to change the name of the .desktop file from
"godot3.desktop" to just "godot.desktop"

What make you believe this should be fixed in godot3 and not in the
implementation of these taskbars?

Doing this would make it harder to introduce godot4 in parallel with
godot3.  As the former now is available from upstream, I guess it is
something to consider.

Could the issue be related to the value StartupWMClass=Godot in the
.desktop file, ref
<URL: https://askubuntu.com/questions/367396/what-does-the-startupwmclass-field-of-a-desktop-file-represent >?

Perhaps the content of misc/dist/linux/org.godotengine.Godot.desktop
need to be adjusted in addition to its name?

The latest packaged version of Godot in unstable (3.6+ds) patches the
StartupWMClass to be Godot3, which may fix this.

Could you try the latest version and see if this behavior still occurs?