I understand the whole "Tell each other apart" and the concept of family names but there has to be more to it right?
More can be said by a new contribution, but until then: u/yodatsracist offers some insight to family names and their relation to states and bureaucracy in their answer to As wives traditionally take their husbands' surnames, does that mean there are fewer surnames than in the past?, particularly in part II of the linked post. I recommend reading James C. Scott's work mentioned and linked in the answer as it's pretty fascinating stuff.