I really don't follow either of these explanations, it seems too hocus-pocus to me as if we are working backwards from the choice..using algebra to divide both sides by a single "factor" (either (x + y) or (x^2 - y^2)) I can obtain all of these choices..I still don't see the proof behind this

That is, if either one is true at a given time, why do they not have to be true all the time? That is, x = y or x = -y. Almost like we are proving one and disproving another and then vice versa to disprove the other and saying they dont have to be true??