Any shape can be hollow or solid. This thread is about solid objects.
Don't take the pizzas too literally. We start with a cone and slice it up. The walls of that pizza are not parallel. They are approximately parallel for thin pizzas. You can then slice the pizza into triangles. As long as each triangle has the same surface area, it will have the same weight. Weight is not mass. Weight is a force. We want to calculate forces, so the gravitational force is weight.
A cylinder is anything with parallel walls, which we don't have. The majority of pools, including kidney shaped pools are cylinders. The base can be any closed 2d surface. But cylinders don't reduce to a point at the origin.
A cone is any closed surface made by pivoting a line through the origin. Mathematically speaking, a pyramid is a type of cone. But you would integrate a straight sided pyramid in cartesian coordinates and elliptical based cones in cylindrical coordinates.
The wiki page for cones covers their definition.