But i'v got one piece in my set of which I can't find the purpose. It's a kind of big dice with all the winds on it and the red and the green dragon also...
I thought it was for showing which was the leading wind of the game, but what are the dragons doing there???
According to this picture in Mah-Jongg's BGG Gallery, it is a wind indicator, which you suspected.
But, I don't know why the Red and Green Dragon symbols would be on there either. Some of the dice shown are blank on those two sides. All I can say is that there are MANY variations of the game, so maybe the two dragons come to into play in one of them.
BTW: My set didn't have a die. It came with four thin plastic disks, each with a 'WIND'.