No, not a thing. As you describe them, these "core" memories would be part of autobiographical memory (memory about oneself). But there is no limited number of particularly special autobiographical memories that define people typically. A possible exception is a highly traumatic event.

Another slightly similar concept is a "flashbulb" memory, where people remember a particular event very clearly (although not always accurately). Classic examples would be the moon landings, Challenger disaster, or 9/11/2001. There are often not autobiographical though, but they might be highly emotional.