Mind clones, computerised versions of your mind and hive minds of a lot of you are an interesting prospect and subject of discussion. "We Are Legion (We Are Bob)" by Dennis E. Taylor is an amusing book on the matter if you wanted such a thing.
As soon as you are not seeing through the same eyes as it were you start diverging, having different experiences and so forth. Or if you prefer identical twins by definition share DNA (we shall leave aside epigenetics for now) and don't necessarily think the same way.
Also part of games is humans are terrible at randomness. If there was a clock in the room you can say "if I look at it and the second hand is a multiple of ? I will do *". Do that as part of a game and things start happening.
Depending upon the level of play you might also trip "yourself" up. I am sure we have not all played every game flawlessly at all points and so there is that. Similarly while I know all the moves of chess and can probably be persuaded to memorise a game as it is happening I am more or less left there -- I never bothered to memorise too many openings, any kind of mid game play strategies or anything else someone that takes chess seriously enough for it to be their thing is something I said "hard pass" on in favour of doing other things*. As such knowing my mind merely means I know my opponent is in the same position as me. I guess if you had some preferences, or better yet some irrational ones, for certain moves, play styles, positions and pieces it could provide something more.
*at this point in time chess is the sort of thing you have to dedicate basically your life to so as to be good (on top of the general affinity for maths and calculation). If others want to do it then I might even find it admirable, definitely not me though.
Thinking about it for chess then I would probably prefer to play an AI than a recent duplicate of myself (or given my opinion above a vaguely similar version of me). I can see it for some games (you better believe bomberman against myself is going to happen).