A customer of mine upgraded their Macbook OS to Catalina, and afterwards, all of their contacts were displaying strangely. In Messages, all of the names were gone and every thread just showed phone numbers instead.
First we tried rebooting. That did not change anything. We checked the connection to iCloud, and it was connected and seemed to be working fine.
What finally fixed it for us was to log completely out of her Apple account and then log back in. After doing this, you have to go through all of the steps to verify the Apple ID password, which required verifications on her laptop as well as on her iPhone on the same account. We made sure iCloud was also properly signed back in. Then, we closed Messages and reopened it. The threads were all properly displaying her contact names again.
Her email also disappeared and that did not come back after logging out and logging back in. However, she uses IMAP, so we were able to just delete her mail accounts and readd them which forced a redownload of the mail and everything was back to the way it had been before the upgrade.