Any straight line through the center point will create 2 identical halves.
Any 2 straight lines through the center point at 90 degrees to each other will create 4 identical pieces.
In both of the above cases each line segment from the center to the edge can be distorted in any way which does not intersect the edge of the square or any of the other lines and the distortion can be rotated 180 degrees in the first case or 90 degrees 3 times in the second to create new identical shapes.
The only special case seems to be using 4 lines to divide the square into 8 pieces which is, I think, the only configuration of 8 pieces.
I think that's every possible solution.
I'm weak at math, but I think that means there are an infinite number of configurations for each of an infinite number of configurations for both of the first 2 solutions, and then there's that 1 extra solution. So does that mean there are uncountably infinite solutions? I'm not sure how you would apply the diagonal method to this.
I disagree, mainly because all of science is based on the idea that this is not true. But I suppose you could try asking someone who explains things better than you.