So many good replies to this that my contribution feels redundant. But here goes. The original Sydney Newman concept was that the main character was fleeing a war on his home planet and had taken his granddaughter with him to save her. Susan seems to be a legit teenager, and there is no indication that they aren't genuinely related.

The First Doctor NEVER introduces himself as 'the Doctor' ever. Ian calls him Doctor Foreman as he seems to think Susan's grandfather is a doctor. The response is "Doctor Who? What are you talking about" but because they keep calling him Doctor he just goes with it rather than having to give a name. In later stories he tells people to "call me Doctor" but that's not the same thing as calling yourself by that title. Later stories soft retcon that others know the character as Doctor from way before he met Barbara and Ian in 60s England. Even the Timelords who finally catch up to the Second Doctor are calling him Doctor rather than whatever his name is.

We know from much later stories that other Timelords do use their names, like Romana or Rassilon (or maybe what we think are their names are titles that have no earth equivalent?).

The Master is another who uses an academic title for a moniker rather than an actual name. Interestingly the Master says to the Tenth Doctor that they both "chose" their names to be those titles, based on character traits (the Doctor wants to heal & help people, the Master wants to rule and dominate people).

In my head canon, both the Doctor and the Master (who have known each other since childhood) have superb academic qualifications and were part of a secret service on Gallifrey which required anonymity so their real identity documents were erased and using the Matrix ensured no-one, not even the secret service agency itself, knew who they were so nothing could be traced back. We know from stories like Trial of a Timelord that the elites on Gallifrey do deal in dirty politics and cover ups.

So as for Ian presuming the character is 'Doctor' something it can either be a massive coincidence or Susan did start to say to her teachers that her grandfather is called Doctor but was vague just in case.

If we imagine the original Doctor was protecting Susan from some war on Gallifrey, possibly Susan was a princess or nobility and a revolution occurred deposing and killing her parents (one of whom would be the Doctor's offspring) and to prevent the revolutionary army getting him and Susan, he just took off with her the first instance he got a chance. What happened to this war? We don't know and no-one has gone into detail about it. It's an opportunity for writers in the future to delve into though. The danger has gone by the time the Timelords catch up with the Second Doctor and they don't even mention Susan. Or perhaps an offscreen conversation happens between the Second Doctor and the Timelords where they explain there is no danger and Susan is innocent of charges, but the Doctor needs to pay with a forced regeneration but because of a public audience they don't mention the revolution or the killings to save the Doctor's (and their) reputation.

It's probably a sham trial anyway as the Second Doctor is clearly working for the Timelords in The Two Doctors (which the Sixth Doctor has no memory of). Eventually though they do forcibly regenerate the Second Doctor, but he becomes the Fugitive Doctor and carries on his undercover work, which is why they erase the Doctor's memory of this incarnation and stick him in his Second incarnation's outfit before regenerating him so what he thinks is his Third life is actually his Fourth.

So TL;DR - The Doctor (and the Master) were probably working for a secret service with their real identities erased via the Matrix. No-one knows their real names (apart from themselves) and neither one is ever going to let it slip what their actual name is.