Is a box a cylinder?

My Calculus 3 professor defines a cylinder as any shape that has identical cross sections for any plane parallel to the base. He used this to explain why he refers to a box as a cylinder. This is confusing, as it muddles the differentiation of solids for me. Is this a common definition?

2. If $X$ is a topological space (such as a rectangle) and $I = [0, 1]$ is the unit interval, one sometimes speaks of $I \times X$ or $X \times I$ as the "cylinder over $X$", see for example the mapping cylinder and suspension.