Yes, x=0 and y= 0 are also solutions, but as you mentioned there are more solutions that later I worked out with a little help. First one has to factor the difference of squares, and the rest is explained below.

x y = (x + y) (x - y) and x y = x + y (given)
Subst. this in the first eqn. gives:
x y = x y (x - y) so x - y = 1 and x = y + 1
Subst. this in: x y = x + y (with x = y + 1) gives the quadratic eqn.: y² - y - 1 = 0 with solutions y = -0.618 or y = 1.618
Knowing that x = y + 1 gives:
x = .382 and y = - 0.618
x = 2.618 and y = 1.618
Seems that both results are solutions of the problem when worked out.