> At least unique enough to use dental records to identify bodies.
Yes but in comparative dental analysis they use ante-mortem dental records to compare with post-mortem remains. It's not like DNA where you can record it once and then use that to match samples decades later in a database. In order to have a high confidence in a match, recent x-rays and records of dental work like fillings, crowns, etc. work best.
And no it is not expected. It's one of the primary challenges with bringing these kinds of drugs to market, as hyperdontia is already relatively common among humans (I had an incisor growing at the roof of my mouth an inch behind my row of teeth). Most successful applications of these tooth regrowth drugs tend to place them near the root of missing teeth hoping that the cellular growth signaling mechanisms are still working.
Teeth wear quite a bit too, I wonder will new ones have to be pared down to fit.