A coin collection is called, well, a coin collection.

The hobby of collecting coins is called numismatics.

a coin collector

Coin collecting

While the president has not said much about coin collecting, he has said a lot about education and about the importance of hobbies that are educational. Both stamp collecting and coin collecting teach about geography and about important historical figures, so the chances are, he would be in favor of such hobbies.

Numismatology