Preheat oven to 350 degrees F (175 degrees C). Spray a fluted tube pan (such as Bundt®) with cooking spray.
Mix whole wheat flour, Greek yogurt, all-purpose flour, baking soda, baking powder, and salt together in a bowl.
Combine brown sugar, eggs, butter, cane sugar, vanilla extract, allspice, and nutmeg in a large bowl; mix until well blended. Stir in flour mixture. Fold in mashed banana and shredded coconut until evenly combined. Spoon batter into the prepared pan.
Bake in preheated oven until a toothpick inserted into the center comes out clean, about 35 minutes. Remove from oven; turn over onto a wire rack to cool, about 30 minutes.
Combine confectioners' sugar, cashew milk, coconut oil, vanilla extract, and salt together in a bowl; stir until mixture is creamy. Drizzle over cooled cake.
