In Methods of Rationality, the Weasley twins refer to the Map as being part of the Hogwarts security system. So it probably gets the information from the wards.

Yes, but how do the wards get people's names? It's not like the name "Ron Weasley" is tattooed on every molecule of the boy's body.

True Names are a feature of some systems of magic — Earthsea comes to mind — but not of the canon Potterverse, nor of MoR as far as I can recall. In canon, the name "Voldemort" has unusual power because of specific spells keyed on it, and it's an adopted name.

Real-time Legilimency? If so, we would expect the map to display whatever name matched a person's self-image; and a sufficiently potent Occlumens could be expected to fool it by sufficiently good Method acting.

On the other hand, there's a subsystem of Hogwarts Castle that does get told the name of every student on their first day, and has a close-up chance to read it from them by what resembles Legilimency: the Sorting Hat. Possibly a similar system is in place for professors and others ... which involves the Headmaster drawing a circle ...