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.