The characteristic may be a number or may have other nature. Changes may be transformations belonging to a group, but not necessarily. One may start from an equivalence relation and the changes would be jumps form an object to an equivalent object.
Invariants play a great role in Physics.
Fundamental quantities, such as energy and angular momentum, are invariants of various physical systems. They are known to be very useful for problem solving. In Mathematics the idea of invariant is also great. However, the traditional Elementary Mathematics does not reveal the whole power of invariants, since it is traditionaly presented as a too static subject, without moves and actions. This is not in the nature of Elementary Mathematics.