If we ask when a movie, game, or book is coming out, "in March" is the best way to say it if you had to choose only one piece of data of the three.

This is only true if both people know you are talking about the future or the past (already released or not released yet) and then implies that the last or next instance of the month is meant. In other words, using just the month only works if the year is already known. Talking about a movie from 2008, the month it released does not give you more information than its year. Using just the month has very limited and short term validity. Which is fine for day to day conversation, but not for written documents or anything else that will be read more than once. In order of the highest information value it's clearly Y, M, D, most significant information to least.