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Do people mean this as flow state? Where you lose sense of time and become immensely engaged with an activity? (e.g. playing an instrument).
To say "I am the game" when the person is a chess player, are they saying they know all the right moves to make? Are they saying they have dedicated 100% of their time to this one game and their lifestyle is 'optimized' for game performance?
Tagged him cause I am thinking Neo from the Matrix movie and descarte is probably most related to that philosophy. I guess plato's allegory of the cave too. will tag both.
Which people say this and when, specifically? They can mean that they are in control of the game, or a million other things. For generic meanings of colloquial expressions English SE is the place to ask.
Your Matrix metaphor could be linked to Baudrillard's famous simulation in the Simulacra, see a recent post in this site which also mentioned Plato's allegory of the cave...