Why doesn't the Marauder's Map account for any transformed animagi's adopted name?
And, how does the Marauder's Map register the names of people in the first place?
As mentioned on this link the Marauder's map works as follows:
The Marauder's Map is a magical document that reveals all of Hogwarts School of Witchcraft and Wizardry. Not only does it show every classroom, every hallway, and every corner of the castle, but it also shows every inch of the grounds, as well as all the secret passages that are hidden within its walls and the location of every person in the grounds, portrayed by a dot. It is also capable of accurately identifying each person, and is not fooled by animagi, polyjuice potions, or invisibility cloaks; even the Hogwarts ghosts are not exempted from this.
Further down you can see that :
Similarly, it identifies animagi by their actual name even when they are in their transformed state. It was by this mechanism that Lupin was able to identify Peter Pettigrew's presence at Hogwarts despite being presumed dead.
Though its not mentioned anywhere how the Marauder's Map registers the names of people, I think it is quite elementary magic to divine name of the people in the world of Harry Potter where you can even remain as a ghost in the world. :)
Purely speculation here, but I guess that it dealt with a person's given name. Names have power, and there are probably other spells as well which deal with a given name. A person's identity may be tied deeply with their given name.
As for your second question -- It's magic. If a person's identity is tied to their given name, then I guess that there must be spells which can find out a person's name.
The Marauders Map uses the Homonculous Charm to keep track of every person in the surrounding area. The charm covered “the caretaker’s cat, Mrs Norris”. It did not distinguish between Barty Crouch Senior and Junior, and Harry did not find out the real name of Peeves the Poltergeist. Importantly, the map knows to use the Animagus names of Lupin, Black, Pettigrew and James Potter.
So the names are not taken from an official register. A cat’s name is unregistered and Pettigrew was believed to be dead. So, the map appears to be taking what people are called by.
So how to distinguish between “Mrs Norris” and say, an ordinary mouse. Well, as the mouse has no name, it’s not put on the map. That would save putting every living creature on the map. Most animals don’t have a name.
This explanation could also solve the alleged plot hole: Why did Fred and George not see that Peter Pettigrew was next to their younger brother all the time. The answer could be that the map displayed “Scabbers”, what he was called, until Sirius Black escaped into the Forbidden Forest, and started calling Scabbers by his real name with sufficient determination, that map then adjusted.