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By the Wallace-Bolyai-Gerwien theorem it suffices to cut the polygon into $n$ sets of equal area, which can certainly be done by continuity of the area on one side of a line as you move the line across the shape. If any of the sets is discontiguous you can rearrange pieces to make it contiguous. Then apply WBG to rearrange each set into the shape of an ...
Every $n$ is possible. As you point out, it suffices to answer the question for triangles. You can divide a triangle $T$ into $n^2$ congruent triangles similar to $T.$ Then these can be partitioned into $n$ sets of $n$ which are congruent as sets and, indeed, have all members congruent. It is interesting to note that, If you triangulate an $m$-gon $P$ into \$...