Let's assume i have created a game, and then created ERC tokens (Example swords) so the player must buy a token to be able to play the game.
What if someone creates similar tokens in his wallet and injects them into my game as if they are bought from my tokens store

Yes your name can be taken and some newbies or friends without crypto knowledge might buy the tokens of that person thinking it's you.
But anyone in the crypto space will know cause they will go through your company's info to check your contract address and check that of his now we that case is solved
I don't even think someone can just wake up and still your game in this way. only if you have no contract address which surely you do.