Position a rack in the middle of the oven and preheat the oven to 350 degrees F. Butter a 9-inch springform pan and set aside.
Combine the brown sugar and 4 tablespoons of the butter in a small saucepan over medium heat and cook, stirring constantly, until the sugar dissolves and the syrup is bubbling. Bring the mixture to a boil and cook until it turns light amber, about 10 minutes. Carefully pour the butter mixture into the prepared pan and arrange the mango on top, overlapping the slices. Sprinkle the shredded coconut on top and set aside.
Whisk together the flour, baking powder and salt in a medium bowl and reserve. Beat together the granulated sugar and remaining 8 tablespoons butter in a large bowl with an electric mixer at high speed until light and fluffy, about 6 minutes. Add the eggs 1 at a time, beating well after each addition. Beat in the vanilla. Add half of the flour mixture and mix at low speed until just combined. Mix in the milk, then add the remaining flour and mix until just combined.
Gently spoon the batter into the pan over the coconut and mango and spread evenly. Bake on the middle rack until golden brown and a tester comes out clean, about 1 hour. Let the cake cool in the pan on a rack for 10 minutes.
Run a thin knife around the inside edge of the pan, then invert a plate over the pan and invert the cake onto the plate. Let the cake cool completely on the plate, at least 20 minutes, then serve at room temperature.