There is no definitive answer to what restrictions there must be; "cylinder" can mean what a given author so chooses. Standard cylinders take $\Omega$ to be a circle (if referring to the surface) or a disk (if referring to the filled in "solid"). The Wikipedia article calls the resulting surface a "generalized cylinder" in the case where $\Omega$ is allowed to be an arbitrary curve, and an "oblique cylinder" if you relax the condition of translating perpendicularly. It also states:
In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively.
I guess I would call your solid cylinder with a hole in it an "annular cylinder", or simply the space between two cylinders with a common axis (if that is the shape you have in mind).