Definition: Two finite sets of polygons A and B are congruent if we can match polygons in A in a one-one manner with polygons in B with each matched pair of polygons mutually congruent.

Question: For every integer n, can every polygon be partitioned into n sets of polygons such that the sets are congruent to one another?

Remarks: For n =2, the answer is yes. Indeed, one triangulates the input m-gon, then cuts each triangle into 3 kite polygons (https://en.wikipedia.org/wiki/Kite_(geometry)) that meet at a common vertex at the incenter of the triangle, thus resulting in a total of ~ 3m kites. Then we cut each kite into 2 mutually congruent triangles and send each triangle into one of the output sets. This approach results in 2 congruent sets of polygons (indeed, triangles) with ~3m elements each. For n=4, one can take the n=2 solution and divide each triangle piece in each set via kites into 2 congruent sets of 3 triangles each thus resulting in 4 mutually congruent sets with ~9m triangles each. This approach should work for all powers of 2 values of n. A natural question here is whether we can manage with less number of pieces in each congruent set.

Guess: For other values of n, including even 3, the answer may be "not always". I have no proof for this.