I have recently updated my MacBook Pro and my partner's MacBook Air to Catalina and have had the issue with the System Preferences icon badged for a update when the software is up to date.
The solution (provided to me by Apple Support) is to use the Terminal command (one line)
sudo defaults delete /Library/Preferences/com.apple.SoftwareUpdate.plist LastRecommendedMajorOSBundleIdentifier
defaults delete com.apple.preferences.softwareupdate LatestMajorOSSeenByUserBundleIdentifier
followed by
System Preferences - Software Update screen, and then press Command + R to force a refresh