I am honestly surprised I hadn't noticed this a while ago. It occurred to me while I working on some stuff involving finding the perpendicular vector to two others in 3 dimensions to form a coordinate frame on a parametric curve (apparently it is differential geometry) that the cross product had the same name as the set theory operator.
Why is this? A cross product to me seems like generating a set consisting of all ordered pairs of two sets whereas the other cross product takes two vectors and returns a third perpendicular one. Was there a reasoning behind this nomenclature? The fact that both deal with non-numerical objects (and a vector triple does form a basis for 3-dimensions) leads me to believe both are either the same thing (in some weird lets-define-vectors-with-set-theory way) or they both represent the same concept for both sets and vectors.
Referring to the set theory cross product by the words "cross product" is technically wrong. It's officially called the "cartesian product." Some people just call it the cross product. In general, confusion should not arise. Ambiguities like this are often tolerated in mathematics, when technically, to be rigorous, such abuse of notation and terminology should be intolerable. In practice, however, it happens, and doesn't really cause problems because everyone knows what is meant. You'll see what I mean when you get into abstract algebra and all of a sudden, 8 different operations are denoted by the same symbol in the same problem just because it's too hard to write it the correct way by hand.