I would like to know if
bring grist to the enemy's mill
play into somebody's hand
are idioms or proverbs and if their meaning are the same?
Bring grist to the enemy's mill would generally be used in the form of an accusation of a teammate or friend to effect of calling them a traitor, double-agent, or saboteur. The guilty party would be doing so for profit or favor from the enemy. Also, the one "bringing the grist" would have done so on their own accord.
Play into somebody's hand would be used when a person is tricked into performing a task by another (usually an adversary). In this case the person performing the action would be unaware that those actions would be causing a benefit for another; or the person asking for the action has a different motive than they expressed.
The word to describe these phrases would be "metaphor" as you are likening one situation to another. The difference between idioms, proverbs, and metaphors can be hard to distinguish.
These phrases would not mean the same thing, but could be used in similar situations depending on context or point-of-view.