When joining a Google Group, how can I make it mandatory for people to leave their first and last name?
You could go to the group Manage page / Settings / Identity
and set 'Required forms of identity' to 'Google profile only'.
Or don't set the group join permission to allow the public to join, but instead only either allow anyone to ask or invite only.
For 'Anyone can ask' , you can enter a 'New member question'. In it , you could tell them to fill a username when joining.
When inviting/adding people to the group, you could specify a display name for each member.