GOOD QUESTION. There is no exact answer. As the scope of work of actual companies using these titles has made the boundaries unclear.
Which is correct, Facilities Management Agency, Facilities Management Company or Facility Management Company?
Facilities Management Company or Facility Management Company In terms of actual work there is relatively no difference between these entities although having said that the scope of work may vary and/or any company/agency can decide to specialise in a particular field. The same as Construction companies can specialise.
Facilities tend to relate to the individual facilities on a site, Air conditioning and heating for example whereas Facility tends to relate to the site as a unit. for example an office block, a Prison or an Airport.
Facility Management could arguably be said to be more comprehensive in scope of work as "They may run the complete site" including cleaning, gardening and security as well as more technical aspects. However, this is not always the case as Facilities Management Companies are often looking to increase their scope and Facility Management companies do not always have complete control of a site as Contracts are often offered in packages.
facility noun (BUILDING) a place, especially including buildings, where a particular activity happens: Cambridge English Dictionary
facilities B1 [ plural ]: the buildings, equipment, and services provided for a particular purpose:
Agencies, (although interchangeable with Companies in some peoples thoughts), tend (in the UK at least) to be more related to "Over sight" of projects and /or Labour (although this is not always the case) A good example would be a New Prison where a Facilities Management Agency possibly working for HMPS or the Main contractor would have Over sight of the Facilities Management Company(ies) actually doing the work. A cleaning Agency would often seem to be just another company but is most likely just coordinating sub contract cleaners and allocating them to clients.
agency noun a business that represents one group of people when dealing with another group: or a government organisation: Cambridge English Dictionary
Examples travel agency.... Central Intelligence Agency