Preheat the oven to 325 degrees F (165 degrees C). Lightly grease a 9-inch springform pan.
Combine coconut, gingersnap cookies, and melted butter in a bowl; mix until evenly moistened. Press cookie mixture into the bottom and slightly up the sides of the prepared pan.
Bake crust in the preheated oven until browned and set, about 10 minutes. Set aside to cool.
Reduce oven heat to 300 degrees F (150 degrees C).
Beat softened cream cheese in a mixing bowl until smooth. With the beater set to medium-low, slowly pour condensed milk into the bowl, mixing only until just blended and stopping to scrape the sides of the bowl as necessary.
Add eggs, one at a time, beating well after each addition; scrape the sides of the bowl as necessary.
Pour about 1/2 of the cream cheese batter into a separate bowl. Stir lime zest and lime juice into portion in the new bowl; pour batter over crust in the springform pan, smoothing into an even layer.
Stir coconut extract through remaining cream cheese batter; pour over lime-flavored batter in the springform pan, smoothing into an even layer.
Bake in the preheated oven until top of cheesecake springs back when gently pressed, about 45 minutes.
Turn oven heat off, but leave cheesecake inside with the oven door slightly ajar until the oven cools completely. Refrigerate until thoroughly chilled.
Prepare mango coulis by puréeing mango with sugar in a blender until smooth. If too thick, add 1 teaspoon water at a time, using just enough to make it pourable. Drizzle over cheesecake when plated.