How to stop Catalina from continually telling me I need to sign in again?
If signing in several times doesn’t resolve this and you can’t or won’t wait it out, signing out completely appears to get this prompt to go away. 


iCloud Keychain refuses to setup on Catalina


You do of course have to deal with the side effects of signing out like choosing to delete or keep local copies of iCloud data and then setting up your iCloud preferences and settings. 

If you have a good backup, you could also try force quitting the system preferences app before you log out of the account.