Make pineapple-mango syrup: Put pineapple cubes, mango slices, sugar, and 2 cups water in a medium saucepan and bring to a boil over high heat, stirring until sugar dissolves. Lower heat to a simmer and cook until pineapple and mango are softened, about 15 minutes. Let cool to room temperature, about 1 hour. Pour into a blender and purée into a thick syrup. Strain through a fine-mesh strainer and chill until ready to use.
Make plum syrup: Put plums, sugar, and 1 cup water in a medium saucepan. Bring to a boil over high heat, stirring until sugar dissolves, then lower heat to a simmer, cover, and cook 20 minutes. Let sit until slightly cooled, about 10 minutes. Pour into a blender with lemon juice and purée; taste and add more lemon juice if you like. Strain through a fine-mesh strainer, adding more water if needed for a thick syrup. Chill until ready to use.
Make shave ice: Process ice cubes in a food processor, 2 cups at a time, until cubes stop clattering and look like snow, with no lumps. Or, use a shave-ice machine.
Add a scoop of vanilla ice cream to each serving dish. Top with shave ice, then drizzle with either syrup plus a drizzle of sweetened condensed milk. Sprinkle with coconut flakes.
Make ahead: Syrups, chilled, up to 5 days. Shave ice, frozen, up to 1 day (whirl again before serving to break up icy clumps).
Per serving with 2 tbsp. Pineapple-Mango Syrup: 361 Cal., 14% (51 Cal.) from fat; 7.2g Protein; 5.6g Fat (3.6g sat.); 71g Carbo (1.6g fiber); 104mg Sodium; 28mg Chol.
Note: Nutritional analysis is per serving with 2 tbsp. Plum Syrup.