equality of games
In combinatorial game theory, equality of games is a defined concept. It certainly doesn't coincide with the idea of having the same game (what you could call an identical copy).
Then two games G and H are by definition equal when G >= H and H >= G.
This is an obvious definition to make, from a mathematical point of view. One also wants to be able to compute with this idea. The theory of canonical forms of games is designed to do that.
Two equal games G and H have the basic property that in a sum of games, one can be replaced by the other, without changing the overall outcome (win or loss).