Actually gifts to the United States, come under the control of the National Archives. They take control or possession of gifts given to the US or to the president.
Tokens and Treasures exhibit features gifts to U.S. presidents by American citizens and foreign heads of states.
www.archives.gov
As the highest representative of the people and government, the President accepts gifts on behalf of the United States of America. The phenomenon, as old as the Presidency itself, grows with each administration: Today a President may receive 15,000 gifts a year. They come from every state in the nation and every country in the world. Gifts from foreign leaders continue a rich diplomatic tradition of exchange between heads of state; those from citizens, both Americans and others, symbolize an inherently democratic exercise - ordinary people freely addressing, in every manner and form, the President of the United States.
The National Archives takes possession.
Like they tried to do with Trumps presidential papers (especially the classified ones) that Trump refused to give up.
See the problem.