Greetings tadé209,
Let’s have you do this. First, please be sure you have a backup of your contacts saved, this way as you’re troubleshooting there isn’t a risk of losing any data. Here’s how to do that, if your contacts are syncing with iCloud - Archive or make copies of the information you store in iCloud
Export contacts on your Mac
1. Open Contacts.
2. Select All Contacts.
3. Click a contact in the Contacts list.
4. Choose File > Export > Export vCard or choose Edit > Select All to select all cards for export.
5. Choose where you want to save the files, then click Save.
If you use lists, consider archiving your entire Contacts or Address Book database as well as exporting vCards. The vCard method above copies your contacts, but not lists. Archives are more complex, but you might not be able to restore data using Archives while connected to the Internet and signed in to iCloud.
To export an archive, open Contacts and choose File > Export > Contacts Archive.
Export contacts from iCloud.com
Export iCloud contact information as a vCard
Export contacts in Outlook on your PC
See Microsoft's instructions for exporting contacts.
Then, pick a contact, remove their birthday. Choose Done, then re-add the birthday. Does that now show up on your iPhone calendar?
The next step, would then be to go to Settings > your name > iCloud > Contacts, uncheck Contacts and then reselect it. This will allow your contacts to sync with your iPhone again.
Thanks.