Assay offices did and do exist in the U.S., but they are affiliated with the governments coinage mints and serve only the governments purposes in that field. They are not involved in hallmarking, as there has never been a hallmarking scheme in the U.S.
In the 1800s, the functions of assay offices in the U.S. included receiving bullion deposits from the public and from mining prospectors in the various American territories. The assay offices that still operate today function solely within national coining system (including bullion coinage for sales to investors).