I looked up set theory a couple of places (WP[1] and Britannica[2]) and didn't find any references of the OR operator in this context.. do you have a link?
The analogue of the OR operator in set theory is the union operator. People think of them as basically the same thing because of the correspondence between a property and the set of things with that property. If A is the property of being either B or C, then the set of the things that are A is the union of the set of things that are B and the set of things that are C.