I think the part that confuses people is this line which comes after cases in which any variable you switch with y or x (keeping in mind y = x) should also accept any variable. But it's a completely different equation.
(x+y)(x-y) = y(x-y)
The math is impossible because the equation is already limited to only one possible outcome, which is x = y = 0, unlike the previous equations where x = y = anything-under-the-sun, and so you need to start paying attention to what your operations do simultaneous to other operations. Here's the kicker, which can be looked at from a different angle other then "Omg divide by zero"
When you perform the elimination to get from the first step to the second step here,
2y = y
(2y / y = y / y)
2 = 1
You are simultaneously assuming y can be substituted as 1 through a similar operation with the exact same result
2y = y
2(1) = (1)
2 = 1
which is impossible as you can see from the inequality and further up the equation chain. It is possible if you eliminate y by substituting it as zero, but other then that an elimination operation like that division breaks the rules by simultaneously stating x = y = 1.