Earlier quoted context omitted.
> They're examples of someone being nicknamed off of their last name instead of their first name? McDonald's, Hardee's. > Caesar isn't even a last name. Worked out fine for Julius.
You're going to have to elaborate. What nickname do you see in "McDonald's" or "Hardee's"? What do you mean by "worked out fine for Julius"? You are aware that Julius Caesar's surname was "Julius", right?
There are modern people today with a surname of Caesar. Somehow, the name of the late celebrity Sid Caesar pops up for me. Wikipedia has more: https://en.m.wikipedia.org/wiki/Caesar_(surname) -- and I know of other languages where their cognate for Caesar is also a surname.
But yes, it is common to address people by their last name as a nickname. I seem to recall it was especially common back in school, something guys did kind of informally. Plenty of reasons explained here: https://www.google.com/search?q=call+someone+by+last+name
I also recall nickname variations of various surnames. Somebody called Fitzgerald might be called Fitz. Somebody called Smith might be called Smitty. Those are two I recall from school days.
And yes clearly, McDonald's and Hardee's are named for surnames.