Wrong name in Family Sharing?
iOS Family Sharing uses Settings->Contacts->My Info
This allows you to choose which of your contacts has information about yourself, such as your name.
Alternatively, this is accessible at the top of the "Contacts" app as "My Card."
I believe one of my kids had edited this 'My Info/My Card' contact. Family sharing is using this to display whatever nickname I prefer on this particular device. Perhaps if the contact was named "me" then I'd see "Me."