Anzu, like so many other people/events in the book, is an early clue to Nick's true identity—one of those clues we skate right over the first time we read.
When Nick summons Anzu at the Goblin Market, Anzu chooses to appear in the form of a man with clear blue eyes. When Nick looks at him, he sees "his own face reflected in those eyes, black eyes and black hair, a face far colder and more grim than the demon's face before him. That was Anzu's intention, of course" (5.106).
And when Anzu speaks Nick's name, he pronounces it "as if it was a rather good joke" (5.106), which—to someone who knows Nick's true origin—we suppose it is. But ultimately the trick is on Anzu, because he, like Black Arthur, is unable to see how Hnikkar has evolved after sixteen years in a human body being shown unconditional love by his devoted brother. Tough luck, Anzu—doesn't look like you'll be getting that human body you were promised anytime soon.