If you have the phone number showing instead of the contact name, try rebuilding your Spotlight index. Even though someone stated above it is an iCloud issue and has nothing to do with Spotlight, it's the only thing that worked for me. I use Google as my Contacts account, so iCloud is not on for Contacts on any of my devices.
I also had an issue where a few of my contacts who are listed only as the first name in my Contacts app (let's say "Dave" as an example) began showing up in the notifications as "Maybe: Dave Lastname". So it was adding the Maybe: and the last name to a contact that doesn't have the last name listed in the card. Weird. For this, what worked for me was turning Handoff off and then back on for both iPhone/iPad and the Macs with the issue. I rebooted both the iOS devices and the Macs. The "Maybe" problem was resolved. Not sure if the reboot is necessary, but that's what I did.
Both of these behaviors began for me after upgrading to Big Sur. In my case I believe it was after the last 11.2.2 update. Affected both of my Macbooks.
Hope this helps someone.