the Doctor, much like John Smith, is simply something that other people have called him and that he's chosen to continue to go by. The Master, on the other hand, deliberately chose the name, though his original name was Koschei, according to the novel Divided Loyalties. The Rani was originally known as Ushas, but presumably she chose her new name as well (as noted in the same novel).
Other Time Lords go by their real names. Rassilon, Romana, Borusa... There are exceptions, but most Time Lords do tend to have names.
Why do humans do the same? Examples are:
- Ace (in Seasons 24-26 of Doctor Who) used her nickname because she hated her given name.
- Minnesota Fats (real life pool player).
- Caligula (real life Roman Emperor and outright nutter). Nickname means "Little Boots".
- Sting (real life musician).
- The Sundance Kid (real life outlaw).