Well, not all children are possessive, in fact, some adults are extremely possessive too. Hmm I guess the question should be why some people are very possissive, without determining their age, gender or background. It's just another human characteristic. It's like asking why some people are calm, moody, intolerant, selfless, etc.. you're questioning a human nature.
I guess the way by which one is brought up affects greatly on these characteristics, society also contributes to the development of one's character, but seriously some people are just born possessive! That's who they are and they will never change! :/