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?
In third-semester calculus, I tend to agree with Rob Arthan: Calling a solid rectangular parallelepiped a "cylinder" (or even a "rectangular cylinder") is unusual (if not rare), and nominally unfriendly to the students, even if doing so is logically defensible.
In differential geometry, one sometimes defines a (generalized) cylinder to be a surface comprising a family of parallel lines, one through each point of some plane curve; see for example B. O'Neill, Elementary Differential Geometry, second edition, p. 146. In this setting, one is careful to speak of a (right) circular cylinder.
O'Neill also writes, "...unless the term generalized is used, we assume that cylinders are over closed curves...." That is, he really does use "cylinder" to denote surfaces with non-circular cross section.
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.