No more than saying "yours truly" means you are literally that person's property. A more modern phrasing of the same sentiment might go something like "happy to be of service, as always".
https://english.stackexchange.com/questions/362307/where-did... has some examples, including a case where a letter ending "your obedient servant" got a reply from "your humble servant." It's a style, a formal way of showing respect through humility, not meant to be taken literally.
What does Obedt mean here? Was Hamilton really literally a servant?