imho i believe that all money that a child receives for their birthday/chanukka/etc… is theirs. I put it all checks or cash over $10 that they recieve in the bank for them for when they grow up. The person that gives the money intends it to be for the child, whether to spend on a gift at that time or to do as they please with it. I know that some parents use their childs money and i dont think its wrong, i just have a different opinion.
As far as using your sons bar mitzvah money to pay for the bar mitzvah, i dont think its fair or, as others have mentioned, i dont feel its right due to his age.
My children are young and sharing is still a skill to be worked at. Whenever they fight over a toy or game i tell them that everything in the house belongs to Totty and Mommy and we share it with all of them. Personal items are a different story.