Huh. That's weird, since she was introduced as Peregrine Mendicant. That's just what I always call her. Sometimes Parcel Mistress if I'm referring to her before she came to Earth. Never heard Purebred Monarch before though. XD I don't even think she is a monarch?
Well, I guess it's spoilers, but she is the Queen of the Exiles. For a while, she was Prospitian Monarch, as she is a Carapacean from Prospit; when she gained Bec's doggy features from donning the ring, the P changed to Purebred.
Their names change with their various forms, but they usually keep their initials... the Mayor is WV, which can stand for Warweary Villein, Wizardly Vassal, or Wastelandic Vindicator And AR is both Aimless Renegade and Authority Regulator. White Queen becomes Windswept Questant and the White King is the Writ Keeper. Same with DD, HB, and CD, they all retain their initials throughout various iterations; the only one who really changes is Jack Noir, who becomes Spades Slick, Bec Noir, or Jack English (depending on which one you're talking about and when).