His name is The Doctor. Just that. The Doctor. That is his name, I believe things where he is from is different, they have names, but not normal ones like "David Tennant" or "Matt Smith" Just a title.
Not all Timelords go by titles. Romana went by her name, but it was so incredibly long that the Doctor just shortened it to Romana since that was the first part of it. And his granddaughter was named Susan (though there's debate about whether she was actually a Time Lady or not, or the possibility that she was part human).
Truthfully....I don't know where they started to call him Dr. Who......I guess supposedly after years and years of time travel he kind of forgot himself and only knew he was a doctor....but not really sure which type of Doctor he is... =/ sweatdrop