The coast here refers to the actual coastline; the narrow area where land and ocean meet. To be "off" the coast means to be some distance away, and typically in the ocean. One would usually use context to determine if the sentence means on land or in the ocean, but you can pick up clues from the word choices. "....off of..." or , "...off the..." is usually used when referring to something that is in, or has occurred in the water near some specific coast or coastal place.

"Off the coast" means "near" or "close to" the coast in this situation. You could also say "just off the coast" which would mean, very close to the coast, or "not far from the coast."

I would never have thought of "off the coast" as being possibly on land. I have only ever seen the term used for in the sea, near the coast. . Possibly a regional difference. The ship sank just 1km off the coast. Off the coast of the northern regions are many islands.