I think there are very few such solutions. The pieces must be identical, and they must touch the center. Consider the segment joining the center with one of the vertices. Then all small figures (in which you split the square) must contain a segment of this length, and there are only four such segments. Any such segment belongs to at most two small figures, and we find that there are at most $8$ small figures. From here on it is easy to see that the possible splits are:

 - the square itself
 - the square cut by a diagonal
 - the square cut by two diagonals
 - the square cut by parallel lines through the center
 - the square cut by parallel lines through the center and by its diagonals
 - the square cut by a line through the center
 - the square cut by two orthogonal lines through its center.
 - the square cut by any smooth curve symmetric by its center.
 - the square cut by any smooth curve symmetric by its center, and the rotate of this curve by $\pi/2$.

There are indeed many solutions. Sorry for my initial remark.