when you take lim as (0, y)--->(0, 0), you're approaching (0,0) along the y-axis.
Likewise, when you take lim as (x, 0)--->(0, 0), you're approaching (0,0) along the x-axis.
However, if you try approaching (0,0) along the line y=x, you have lim as (x,x)--->(0,0) = (8x^2*x^2)/(x^4+x^4) = 8x^4/2x^4 = 4.
Thus the limit is path dependent and does not exist. You can see this visually by looking at the graph of the equation in 3 dimensions here:
You can clearly see the limit is 0 along the two axes and 4 along the line y=x.