I completely think it’s mostly just a shorthand, gym teacher thing and the switch from purely last name is to differentiate. (I hate being called solely by my last name like gym teachers always do.)
Clinton, Bush, and even Sanders can conjure multiple persons to mind. (How many ‘Muricans would have thought of a fried chicken mascot before a politician back at the beginning of his campaign?)
We also tend to do that with the Kennedys, Roosevelts, and other celebs like the Jacksons and Baldwins.