Preheat the oven to 350 degrees F (175 degrees C). Butter two 9-inch cake pans and sprinkle with flour.
Mix flour, baking powder, and salt; set aside.
Beat white sugar and butter together in a bowl using an electric mixer until very light and fluffy, about 3 minutes. Add eggs one at a time, mixing well after each addition. Beat in vanilla extract. Add flour mixture in batches, alternating with milk, beating batter briefly after each addition until thick and smooth.
Divide batter equally between the 2 prepared cake pans.
Bake in the preheated oven until a toothpick inserted into the centers comes out clean and edges are lightly browned and lift away from the sides, 25 to 35 minutes.
Beat butter in another bowl until fluffy. Add sugar 1 cup at a time, beating well after each addition. Add vanilla extract and mix in milk until frosting reaches your desired consistency. Do not refrigerate frosting before spreading on the cake.
Let cakes cool to around room temperature, at least 30 minutes. Remove from pans and refrigerate until cold and firm, about 20 minutes.
Place 1 layer of cake on a platter. Spread 1 inch of frosting on top, avoiding the sides. Push about 1/3 of the strawberries and blueberries into the frosting. Top with the second layer and repeat with frosting and adding berries. Top cake with another inch of frosting and the rest of the berries. Refrigerate for 30 minutes before serving.