Hi Boyd, good question. If I'm not mistaken, the Office of Production Management officials would meet with manufacturers in order to acquire new weapons and contracts. Again, if I'm not mistaken, the officials included representatives from the Army, the Secretary of War, and the Secretary of the Navy. They would be responsible for securing the production and supply of arms and equipment for the nation.
So, if I interpret that correctly, that would make those particular flamethrowers, acquired by the Navy, used by the Marines, never the less, a Navy piece of equipment.
HERE is an article about the Office of Production Management.
Hope that helps a bit.