MacOS is showing a badge notification on System Preferences. How can I make it disappear? The badge notification is because it wants me to upgrade to Catalina. I never want to upgrade to Catalina because my hardware is too old to be run under any newer version of MacOS than I already have.

How can I dismiss this particular notification or otherwise make the notification badge go away?

I tried System Preferences --> Notifications, to see if I could disable all badge notifications from the System Preferences app, but that doesn't appear to be an option. And really, I don't want to disable all notifications from System Preferences, I just want to tell it to to stop nagging me about the Catalina upgrade.
I had the same complaint. I removed System Preferences from the Dock, and and haven't been bothered by the badge nag.

System Preferences is still easily available in the Apple Menu > System Preferences.

And you might review this Stack Exchange question for an alternative solution.