The "many faced god" can refer to more than one thing. The term is used both a metaphor, and as a reference to an ostensibly real deity.
There is idea that death has many "faces" in the sense of ways to die.
There is the idea that every culture has a god of death, whether it be the Stranger, the Black Goat of Qohor, the Lion of Night in Yi Ti. The House of Black and White is filled with statues of the different representations of the god of death.
There is the idea that followers of the god, the assassins who carry out it's will, are legion. This connotes a sense of anonymity that is a source of their effectiveness.
It is not clear at what point they gained the magic to actually change faces, but this is not an aspect mentioned in the foundation of the cult.
Martin does go into a fair amount of detail on the origin of the cult in the slave mines of Valyria. Beyond that, there is not much concrete material.
It is unlikely that the "Waif" and the "Kindly Man" are the same person. In the books Arya sees several other faceless men in the House of Black and White. In the books, the Kindly Man is not revealed as Jaqen H'ghar, who may then be yet another faceless man.
Additionally, in the books there is no indication that their magic extends beyond the ability to change faces, thus, no reason to assume was is all an illusion, but the show has complicated this by depicting scenes that were surely illusions. (i.e On the show, the faceless men seem to have powers similar to the warlocks of Qarth.)