Okay, I just checked again, and I found that the Delta Flyer shuttle does have a registry number on it, but no name. Also, the registry number doesn't change when you change it in the game.
I named mine 'U.S.S. Lady-1' (because my ship is the U.S.S. Lady, named after my beloved dog), and I changed the registry to 'NX-92509-A' (same as my Intrepid, with just the -A added, since we can't add -1 or -01 to the registry number). However, when I looked at it in the ship tailor, it only shows '74652' on it (no NX; also, no name that I could see), which is what it was originally, before I changed it. I hope that it gets fixed soon. (Perhaps we could have caught this bug if it had been up for testing on Tribble... )
Ticket ID #1,044,897
(Otherwise, I love the shuttle, and the pet shuttle that comes with it. Thanks for putting it in the game, Cryptic! )