Is Japanese kasutera the same foodstuff as Korean kastera?
There is a particular baked good called kastera in Korea, but they pretty much use the term as a generic name for any Western-style sweet cakes (source: my Korean mother-in-law). So the answer is yes and no. You will find a kastera equivalent to Japanese kasutera, but it is a subset of the kastera set.

The closest comparison I can think of would be the term "barbecue." It would very specifically mean slow-smoked meat in certain regions of the United States. In other regions, "barbecue" is a bigger tent, and includes, among other things, anything that is cooked on a grill.