Here is the trait in question:
TALES OF THE TMNT: At the beginning of the game, for all characters that can use this trait, choose a "Tale:" trait that a friendly character can use. This game, all friendly characters with a "Tale:" trait can use the chosen trait instead of any other "Tale:" trait.

If I were to KO the turtle that my opponent chose for their Tales trait, would that mean the remaining turtles could no longer use it? What if I mind control them?

It says "this game" so the other characters can use it for the duration of the game even if the character they took it from is K.O'd

Pg. 28 in the rulebook, under "Durations"-