So, there’s this TV show, right? About a character called Doctor Who.
You can tell it’s about a character called “Doctor Who” because the show is called “Doctor Who” and in the credits for the first eighteen years or so, the character is called “Doctor Who”. (Or sometimes Dr. Who. But I think we can safely put that in the same conceptual space.)
Some people think he’s called The Doctor, and calling him Doctor Who is incorrect.
These people are wrong.
Analogy:
There’s this character called Robin Hood.
He’s had many stories told about him, over hundreds of years.
And yet, somehow, relatively few people manage to get their knickers in a twist that his given name is Robin of Loxley. (Or, occasionally, Robert, Earl of Huntingdon.)
And no-one feels obliged to point out that his last name is not “Hood” and that therefore calling him that is somehow wrong, or silly, or incorrect.
So. Doctor Who it is then.
Because it’s his name.