Is damsel in distress an idiom?

No; it comes from the French. The Word "damsel" (an old Word that used to refer to a 'young lady') is not used in conversation any more. The expression means a lady in need of being rescued, and is a common plot in Fairy Tales and old-fashioned adventure stories.

An idiom is a phrase that makes no sense unless you know the definition. This is an ordinary phrase that means exactly what it says.