If the cake is only iced on the sides, you could cut it through the center the way that one would cut a cake for layered cake, and you would still have the same amount of icing on all sides. Do this 8 times, and you'll have 9 thin pieces of cake all the with the same amount of icing on the perimeter.
I guess the answer you're looking for involves cutting the cake at weird angles. But suppose your cake is cut into a 3x3 grid to make 9 pieces and that normally, a corner piece has 3 units of icing (enough for 3 sides), a side slice has 2 units of icing, and the center slice has one unit of icing (just the top). Well, why not just ice the cake unevenly so that the center has three units of icing piled onto the top, the sides have three units of icing spread over the two external sides, and the corner pieces have their usual three units spread at a normal thickness? Then you achieve having a square cake with icing on the sides cut into 9 pieces with equal amount of cake and frosting.
Of course, this assumes your cake is actually a cube for ease of explanation, but you can adjust the amount of frosting so that it's equal no matter what the height of the cake.