Is a mother taking from her son's saved Eid money and similar funds considered a debt upon her that must be repaid, or is it a permissible action that does not require restitution?

The basic principle is that whatever is gifted to children becomes their property, and no one is allowed to dispose of it without their permission if they are mature, or in a way that does not benefit them if they are not mature. An exception is made for the father: he is permitted to take from his son's wealth, without his permission, an amount sufficient for his needs for maintenance, and this is not considered a debt. It should be noted that celebrating birthdays is not permissible.