To make the cake, position racks in the center and bottom third of the oven and preheat to 350°. Lightly butter two 9" round cake pans, sprinkle evenly with flour and tap out the excess. (If you wish, butter the pans, line the bottoms with rounds of parchment paper, then flour the pans and tap out the excess.)
Sift the flour, sugar, baking soda, cinnamon and salt into a bowl. In another bowl, stir or whisk the bananas, pineapple, oil, eggs and vanilla until combined. Do not use an electric mixer. Pour into the dry mixture and fold together with a large spatula just until smooth. Do not beat. Fold in the pecans. Spread evenly into the pans.
Bake until the cake springs back when pressed in the center, 30 to 35 minutes. Transfer the cakes to wire racks and cool for 10 minutes. Invert the cakes onto the racks (remove the parchment paper now if using). Turn right side up and cool completely.
To make the icing: Using an electric mixer on high speed, beat the cream cheese and butter in a large bowl until combined. On low speed, gradually beat in the sugar, then the vanilla, to make a smooth icing.
Place 1 cake layer, upside down on a serving platter. Spread with about 2/3 cup of the icing. Top with the second layer, right side up. Spread the reaming icing over the top and sides of the cake. The cake can be prepared up to 1 day ahead and stored, uncovered in the refrigerator. Let stand at room temperature 1 hour before serving.