Thanks, that helps a lot. I've never heard this term used in the United States (except in reference to the horse). Here, "secretary" can be used to refer to both the person and, less frequently, to the office.
Many countries have official administrative agencies called secretariats. It's not a term we use much in the USA, but it's common around the world. For example, the United Nations Secretariat is one of the UN's major offices.