Well, broken comes from game-breaking, making something unplayable because the game could be mechanically "broken" in the purest form of the definition.
Nothing in League of Legends is technically broken unless it's a bug that literally changes up the entire game into something literally unplayable.
But players exaggerate something hard to counter as broken. Ultimately, nothing is unplayable because everything has counterplay, as in every mechanic in the game has some form to nullify it's effects. But when an ability or attribute of a certain champion has little counterplay, that is when gameplay becomes unhealthy and thus players will say it is mechanically, technically and perhaps literally "breaking the game".
So, all of the points above can potentially be hard to counter depending on the champion's specific mechanics so all of those can potentially be "game breaking".