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Then the cross-section is a circle or a point.
If it a right cone then it is a circle, otherwise an ellipse.
A right circular cone is perfectly balanced on its circular base. Imagine a cone that has a circular base, but leans to one side - this is a non right circular cone.
It is a section of a right circular cone by a plane that is parallel to one generating line of the cone.
By definition, the circular cross-section of a cone changes linearly in width as you go along its axis. By definition, the cross-section of a prism is constant along its axis. So, by definition, a cone prism is an impossible shape.