Hi murmur11000,
Thank you for using Apple Support Communities. I understand you're having some difficulties with the Birthdays calendar within macOS Catalina. I'd be happy to help.
If your issue is only happening on your Mac and not your iOS devices that would imply the issue may be with the software on your Mac. Before we continue further do you have a back up? If not that's alright. It's always recommended to have a back up before troubleshooting potential issues. If you'd like to make a back up here is a great resource: Back up your Mac with Time Machine
After you have a recent back up your next step would be to see if the behaviour is replicated in a test user account. Here is a great article on using a test user to isolate issues: How to test an issue in another user account on your Mac
Here's a good resource on how the Birthdays calendar functions:
Show or hide the Birthdays calendar on Mac
Cheers.