The story draws heavily on the history of the Rosicrucian Order
(or Brotherhood, or Society), in which both Borges and Umberto Eco
show a repeated interest in their writings.
In brief, the history of the Rosy Cross started with an anonymous pamphlet, appearing out of nowhere towards the end of the Renaissance and calling for spiritual regeneration, political house cleaning, redistribution of wealth and above all complete and utter anonimity and secrecy on the part of its members. For a good summary of that and subsequent publications see the excellent node on the subject.
It was claimed at the time, and is pretty much accepted as fact today, that the whole Rosicrucian thing was an elaborate prank. Nevertheless the two original pamphlets led to a deluge of supportive publication, outraged criticism and any amount of speculation re who these people were and how they were going to bring about the social revolutions they were preaching from the shadows of anonimity. Very much in the same way that Tlon transmogrified from a joke to a popular "fact", so the Rosicrucian Order became the conspiracy theory that wouldn't die - traces of it are claimed to be found in modern day Masonry and shades of it can certainly be seen in the myth of the Illuminati. Borges is essentially parodying the human capacity for believing the unbelievable, as long as it makes sense. Paradox? Borges...
Dan Brown is heavily, if probably unknowingly, indebted to this rather fogeyish little scholarly joke: after the phenomenal success of The Da Vinci Code, he could not resist the temptation to cite his inspirations as "factual sources", thereby giving rise - and credence - to a whole industry dedicated to retroactively "proving" the sources right, thereby justifying the factual integrity of a work of fiction. Remarkable.