Ultimately I think Memor-X has the right answer, but I think there are some other relevant details that are worth remembering.
The Throne of Heroes exists outside of time and space, and the "EMIYA" in the throne has access to all events he has experienced during his summonings in all timelines, worlds, and realities. But normally a summoned Servant has more restricted access, coming only with those memorizes that the hero has internalized (and the knowledge the nature of the summoning has granted him, such as the Greater Grail System in Fuyuki). The "memories" of all summonings contained in the throne are likened to books: they're not internal to his spirit, but can be accessed much like you can just take a book off of a shelf and read it to learn what's inside it.
Now, there's something very special about Archer that interacts differently from all of this: he has a Reality Marble. These exist essentially outside the Throne, and is attached to the heroic spirit as he exists within the Throne. All summoned versions of him access this exact same reality marble. And his reality marble stores all weapons (and potentially other things, like Rho Aias) he encounters. So in order for FSN Archer to be able to project Kanshou and Byakuya (or Rho Aias, etc.) he needs only for some version of himself in some world's timeline to encounter it, and now they all have access to it, regardless of apparent chronology.
In reality things happen more like Memor-X says: rather than witnessing K&B in some fight, Archer is specifically said to have learned of these weapons which were said to be of Noble Phantasm quality but yet no hero was associated with them. These "fit" him better, as the best projections require you to project the skills and abilities of their wielders. The lack of such made K&B easier and simpler to create, as well as sort of fit the theme that EMIYA was "nobody", making them feel more natural and like they were "his".
As an aside, this reality marble is also essential to EMIYA's perspective and spirit warping, leading him to come up with the idea to try to kill (an earlier version of) himself in order to end his own existence. A Reality Marble is the user's inner world made manifest, and the experiences of his summonings impacted his reality marble and effectively carried on their memories to all versions, including the "true" version enshrined in the throne. You can see this by noting how there are significant differences between Shirou's and Archer's uses of Unlimited Blade Works. Archer's looks more apocalyptic and miserable, with a cloudy, ashen sky and large gears. Shirou's tends to have a more hopeful, brighter, bluer sky and far fewer (to no) gears.