I think the factorization is a reference to from quantum field theory. I haven't learned quantum field theory though, so I can't comment much. From what I can gather, multiplying something by the creation operator gets you the same state but with an extra particle.

I can tell you that at the very minimum, assuming Copenhagen and the minimal amount of physics to allow entanglement to happen at all, whenever two of the same kind of particle are entangled, they have a 50% chance of swapping. If you use MWI, it's that I can find a universe with the same probability density in which those particles are swapped.