No individuals, astronauts and presidents included, were ever gifted a moon rock. The only lunar material given away by the United States were the displays prepared using Apollo 11 and Apollo 17 samples.The astronauts were awarded moon rocks in name only as Ambassadors of Exploration, but those samples remain the property of NASA and are on long-term loan to museums and educational institutions.
The majority of moon rocks are held today in vaults in the Lunar Sample Laboratory at NASA's Johnson Space Center in Houston (not a museum). It has no where near the security of Fort Knox (nor is it needed) but it is on federal property, requiring proper accreditation to access.
A smaller cache of lunar material is held at the White Sands Test Facility in New Mexico to protect against the unlikely scenario that something were to happen to Johnson Space Center (e.g. hurricane damage preventing access).
Scientists and museums can request loaned samples for scientific study or public display through NASA's Astromaterials Acquisition and Curation Office.
Harrison Schmitt is currently working on a multi-volume series about the scientific outcome of the Apollo 17 mission, which will include details as to what was learned from the moon rocks that Cernan and he returned.
The lunar samples brought back by all six missions were cataloged by NASA.