I don't think that the heat shield is the same as the shuttle's. The tiles are tougher and attached in a different way, and the underlying steel structure can stand more heat than the shuttle's aluminum could.
Orbital refueling is required for HLS and depends on rapid turnaround so SpaceX and NASA must have some confidence that it can be achieved without a radical redesign.
There are differences from the shuttle's tiles, but they are much more similar than different.
Rapid turnaround seems at this stage more an aspiration than a designed feature. It was the same with the shuttle. In that case it was never achieved at all, even with decades of experience.
Turnaround and reuse are desirable, particularly to minimize cost, but not strictly necessary for a successful mission.
The tiles on Starship are much more uniform than on the Space Shuttle, so they are cheaper to produce and less complex to replace.
(Superheavy is a different story.)
All this turns on the design of the heat-shield tiles. Those we have seen are basically the same as on the shuttle. Each individual tile needs a great deal of personal attention before and after each use, including squirting in some goo that will boil off on the next flight. Many will need to be replaced, each flight.
So, turn-around time is weeks, minimum. More probably a month or two, or three.
It is conceivable that a future version of Starship will have a heat shield that needs less attention. Maybe a fabric shroud in place of tiles, over most of the hull, will turn out to be practical. If that happens, it might get possible to turn around a re-entered Starship faster. But that will require entirely new development not obviously begun.
I will be happy to be proved wrong.
Despite all of the above, even without quick turnaround or even re-usability, Starship is obviously overwhelmingly superior in every conceivable way to Artemis/Ares. NASA has the unenviable chore of trying to unload that turkey on some commercial contractor to get out from under the obligation to pour ever more treasure down its gaping maw, endlessly. If NASA succeeds in selling it at a penny on the dollar to Marietta, and leaving Marietta to try to find someone, anyone willing to pay for even a single hull, we can all breathe a sigh of relief.