The whole "Holmes is a real person" thing is a very carefully maintained charade, including several residences that are empty but given the semblance of activity and sometimes used for meetings. Part of the point here is that any enemies "Holmes" makes will try to attack a nonexistent person. Firebomb the apartment? Well, "Holmes" wasn't in! He was totally in New York at the time!
Moriarty is also an organization, made of rogue Holmes operatives who basically went "we can get paid way better if we use our skills for crime". They use roughly the same tricks to make it appear that Moriarty is a single person, made somewhat easier by him not needing to put in any public appearances. Holmes and Moriarty are of course aware of each other's non-person status, but can't disclose it without blowing their own cover. So while they are mortal enemies in all other regards, they keep mum on that rather salient piece of information!