Because the cloak IS grey. The ones provided for the movies were grey, and color adjusted for certain scenes. But grey is the accurate color.
In the books, the cloak is grey, but can take on the color of green, blue, etc during different times of the day/light and based on the environment. It's meant to protect and hide the wearer, hence the point of giving it to the fellowship.
In the films, you tend to see more green on the three hunters scenes because they are surrounded by the plains of rohan a lot, and trees, etc. The cloak therefore would adjust to surroundings to camouflage them. When it comes to frodo, you see it as grey a lot of times, especially during the scenes where he is near rock, and darkness towards minas morgul.
So there is your why.
But as for why they chose grey as the color to go with for this? It's the color the real cloaks were. They were made from wool. It's a bluish grey wool.
