iCalendar is unable to send invitation

I have an invitation in the iCalendar. I want to add participants. I'm entering their email and see the popup saying that "Calendar will be unable to send the invitation." and the text saying that "To be able to send invitations, you need to create an address card for yourself in Contacts." and proposes to Open Contacts.

But, when I open Contacts, select myself, and click Card => Make this as my card, nothing changes. I tried restart iCalendar after it - still the same issue. Please advise how this weird and annoying issue can be fixed.

Thank you!