elementary relies on applications notifying the OS that they want to use notifications by setting a property in their .desktop file. Applications not built/controlled by elementary are not guaranteed to do this
You can add the following line to the bottom of the .desktop file for the application in question and it should show up in the notifications settings so you can configure it like the elementary applications.
X-GNOME-UsesNotifications=true
For example, my /usr/share/applications/spotify.desktop file looks like this:
[Desktop Entry]
Name=Spotify
GenericName=Music Player
Comment=Spotify streaming music client
Icon=spotify-client
Exec=spotify %U
TryExec=spotify
Terminal=false
Type=Application
Categories=Audio;Music;Player;AudioVideo;
MimeType=x-scheme-handler/spotify;
X-GNOME-UsesNotifications=true
And my notification settings now have the Spotify application as a configurable notifications source:
This is not guaranteed to be a permanent fix, as the application in question may overwrite its .desktop file during updates, etc...