Is there a mathematical designation for a cuboid that has one or two of its edges with infinite length (in essence, forming an infinite subset of the 3-D space)?
This would be some kind of 3-D strip or the band if one edge only is of infinite length and a slice if two are.
One way to call such geometric bodies, including the original finite sided cuboid, is as "polyhedra" if they can be bounded or unbounded, and "polytopes" if they are bounded. Generally speaking, a polyhedron is the set of solutions to a system of linear equations and/or inequalities. If it's bounded then it's a polytope. For a "cuboid" you're talking about, bounded or not, you have linear inequalities given by upper and lower bounds on each axis for each bounded axis.