I've never even heard of a Damascus "finish". All the Damascus knives and lights I've ever seen, are made with folded steel either called Damascus or damasteel (both are similar but different forms of steel in the Damascus. One is carbon and the other is stainless). But no idea what or where the term "Damascus finish" came from.
That might also refer to the fact that Damascus blades were often acid etched to bring out the typical banded pattern.