Stewart began as a job: the steward who managed a household or estate. Old English stigweard joins "house(hold)" and "guardian." After the Norman Conquest the title covered manorial managers; in Scotland stewards administered crown lands, and the royal Stewart (later Stuart) line made the office famous.
Most American Stewarts inherit the occupational surname, not a claim to the throne. Stuart is a French-influenced spelling of the same name. Type: occupation/status office. Families carried the name across the Atlantic without changing its core sense.