I've never read any account of an occurence like this between them, and I wasn't sure what the intendedeffect was supposed to be. For me, it felt a little too contrived, but for some reason I laugh when I hear Wyatt say the line about calling him by one name or the other, not both.
shareBest shot at this...
The "call me Wyatt or Earp" exchange happens soon after he asks Wyatt if he believes in friendship(Yes) and does he have many friends(NO) so when Wyatt asks him to pick one, he chooses Wyatt and offers his hand says "Call me Doc." To which I always took the meaning to be, he is choosing to be friendly. Addressing him by his first name, offering his hand.
WYATT EARP= Standoffish
WYATT= Familiar and friendly.
~~~"Who do you think you're dealing with? Guess again."~~~
I think thats a real good point. I also think it might have been a sort of celebrity-mocking move. Calling him by both names as if his name is so important; maybe chiding more than mocking, and showing Doc wasnt trying to be anyone's friend. All of that would really underscore your thoughts, too, showing how by the end of the conversation they were choosing to be on friendly terms.
shareIt could be as simple as a stylistic choice (i.e. it sounded cool).
In the novel THE GLASS KEY, Dashiell Hammett only refers to the protagonist by his full name; I think a pronoun is never used.
In the novel BARONNE STREET, one of the characters always calls the protagonist by his full name.