I'm not sure. Given that there is an individual standing in front of you, does it ever make sense to use the odds of a random individual instead of your best estimate for that particular individual? At the least, it seems likely that the circumstance in which the name is being given would change things greatly: a police officer stopping someone in the dark versus a minister introducing themselves after a sermon. In neither case does a universal estimate of lying about a name seem applicable.

And the choice of name does seem important. If someone with no visible appearance of being Asian gives you a very Asian name (or vice versa), you might have a lot more doubt. And if the name is otherwise humorous (Biggus Dickus) or stereotypical (John Doe) this also should affect your estimate. Why would it be a plus that "the posterior then doesn’t really depend a lot on what name he says"?