I have been running into issues with "make this my card" for several releases of macOS at this point, up to and including 27 dev beta 4 (Golden Gate). I'm not quite sure what the root issue is but the end result is "My Card" can't be set and there is no error.
I generally use Fastmail for contacts, and only Fastmail, but found through experimentation that if I turned on iCloud contact syncing as well, and had my own contact in iCloud it started working, I was able to set "My Card" properly. When I set up my new Mac things broke again.
I thought at first just having a contact card on iCloud was enough. When things are working properly you can get to your own contact card by going to "System Settings"->"Users and Groups" then clicking the "Open Contact Card..." button for your user account. When it's broken that same procedure will create a new contact card (local or iCloud depending on whether iCloud is set up or not) with the first name set as the Mac account username and set it as "My Card". On my new Mac that kept happening and it would never open the proper card.
One option I tried in the past was a simple AppleScript that sets the card. That works but I ran into other issues as a result of that so ultimately I abandoned that fix.
I found another solution today via stackexchange.com. The user ended up going to iCloud.com and set "My Card" there. This worked for me as well. That is from 14 years ago, this is still broken, and that fix still works!