This meaning refers to a light warm brown shade resembling the coat of a young deer.
She chose a coat in fawn.
The walls were painted a soft fawn color.
As an adjective, fawn describes something colored in a soft pale brown shade.
He wore fawn trousers and a white shirt.
The dog has a fawn coat.
Usually used with over or on, this meaning refers to behaving in an overly admiring or submissive way to gain favor.
Some employees fawn over the director.
He fawned on wealthy guests to gain their support.