In particular, what's the evidence that points to their identification?
If you deep dig enough this Doctor pops up all over the place: political diaries, conspiracy theories, even ghost stories. No first name, no last name, just The Doctor. And the title seems to be passed down from father to son, like some sort of inheritance. The Doctor is a legend woven throughout history. When disaster comes, he's there and he brings a storm in his wake. He has just one constant companion: Death.
They were not identified as "Doctor Who." They were identified as "The Doctor." Doctor Who was the name of the various series that told their tales.
The answer to why he's called "Doctor Who" is actually quite interesting. He was approached by a publisher and asked whether he'd like to write some factual books for children, which ended up being published under the titles of:
Originally the books were supposed to be entitled "The Doctor, who Discovers Prehistoric Monsters", etc., but the cover designer missed the word "The" and designed a logo that treated the author as though his name was "Doctor Who" and the name stuck.
Source: (x)
I believe that answers your question.
There are a few hints. Most of them carry some sort of a magic wand, that over time has been referred to as a sonic screwdriver, despite witnesses confirming that it is neither sonic nor a screwdriver. Its origin, functionality and design vary in each tale, but it is a consistent tool of the figures. The other, simpler way to recognise them is by their transportation method; they tend to always arrive in what has been described as a "magical blue box" that's "bigger on the inside" (Churchill, 1943; Nixon; 1968) and comes and goes with a deafening sound. Portraits of this box have been found all around the world, in caves, Aztec temples, monasteries and temples of all sorts. Its most detailed description comes from the Roman poet Lucius Caecilius Iucundus, alleged survivor of Pompeii, ^^[citation ^^needed] who describes it as:
"The temple of salvation, sent to me by Jupiter in times of desperation; covered in gold and rings of light, with the fountain of life in its centre. I could not comprehend why they had sent their herald to save me and my household, but it was not my fate to know. Only I saw his face, with ice and fire of destruction, but over all, with repentance." ^^[1]
Their means of transportation are all we know to relate them, and it is yet unclear if they are or not the same character, though some members of Scotland Yard report having overheard confirmation during his work with the anonymous detective only know by history as Madame Vashtra, ^^[2] albeit some scholars debate over the veracity of this testimony.
One revisionist interpretation that I've heard is that "The Doctor" isn't really a single person, but an office which is believed by his followers the "Whovians" to be regularly reincarnated via a mystical process. This form of political power via claims of reincarnation in not unknown in other parts of the multivariate.
Do people just like to fuck with this sub? I don't get it.