Butter the sides of an 8-inch springform pan. Cut a 26-by-7-inch piece of parchment paper and press the parchment strip around the inside of the pan so it extends above the rim (this will hold the cake in place as you build it up). Lock the springform ring to secure the paper to the base of the pan.
Pulse the graham crackers in a food processor until finely ground. Add the melted butter and pulse until combined. Press the mixture into the bottom of the prepared pan and freeze until firm, about 10 minutes.
Meanwhile, cut the carton of mango ice cream in half with a serrated knife. Return one half to the freezer and let the other half sit at room temperature, 5 minutes. Transfer to a stand mixer and beat with the paddle attachment on medium speed until smooth and spreadable but not melted. Spread the mango ice cream evenly onto the crust with a small offset spatula. Freeze until firm, 30 minutes to 1 hour.
Repeat Step 3 with the black raspberry ice cream (half the carton) and freeze. (Rinse out the mixer bowl between each flavor.) When firm, repeat with strawberry ice cream. When firm, beat all of the vanilla ice cream and spread on top; freeze.
Repeat the layers with the remaining mango, black raspberry and strawberry ice cream, making sure each layer is firm before adding the next. Freeze the cake until completely firm, at least 1 hour.
Remove the springform ring and parchment. Beat the heavy cream with the confectioners' sugar in a large bowl with a mixer until stiff peaks form. Spread on top of the cake and return to the freezer until firm, about 10 minutes.
To make the ice cream easier to spread, we beat it in a stand mixer. If you don't have a mixer, just let the ice cream soften at room temperature.