Here's my take after 30 years of watching Doctor Who. This isn't based on any single particular source and based mostly on the TV shows and movie.
First, it's a personal choice on the part of the Time Lord, which can change during their various life spans. Time Lord culture seems to expect this as a normal occurrence. For example, see how Drax reacts when he meets the Doctor in "The Armageddon Factor".
Second, what we hear as humans isn't necessarily what the Time Lords are actually saying. Time Lord communication is at least partially telepathic and possibly machine enhanced. Our semi-evolved simian brains are making the best of the data a Time Lord brain sends to us. For example, see how Sarah Jane Smith questions how she can understand every alien she encounters exactly as though they were speaking English.
Third, using Romana's name as a example, Time Lord personal names seem to be very complicated and long, so they shorten them or replace them with titles for us simians to use.
Fourth, out-of-universe it allows authors to create any kind of Time Lord they need for a story and immediately give them something instantly identifiable and iconographic.
Fifth, (extra point shot attempt) "Doctor Who?" was the punchline of many of the early episodes. When the Doctor left the final scene, one of the characters remaining would say something like: "He was the Doctor", and another would ask "The Doctor? Doctor Who?" Asking if anyone had actually heard his name.