Without seeing the inside of the hole it can be hard to tell. But based on the rock, I think it's natural.
The inside of a drilled hole will usually have some particular marks depending on how it was drilled, but by and large it should be relatively smooth and any high points should be worn down. Natural holes can result from lots of different processes, but in the case of a fossil void or crystal pocket the insides usually have some "texture" that would probably be worn or smoothed by the drilling action. (It's not the same process but think about a drilled hole in wood, or a bullet hole in wood.)