Damask is a type of “rich patterned fabric”, and it can also refer to “the pink color of the damask rose”. The word damask is also an adjective (“resembling damask”) and a verb (“to decorate or weave with rich patterns”).
The cloth was named after the Syrian city Damascus, “famous in medieval times for steel and silk” (OED). The word seems like it lends itself to more general uses – maybe to describe anything richly decorated. But a search for damask on Google News shows that it’s almost always used to refer to that type of fabric.