Do cats know it's game
 

Or do they think the string is an animal they wish to murder?

Cats can play some games but "chase" games (string, laser pen etc) is mostly their hunting instincts kicking in... It's not even that they think it's a real animal, they know it's string, but they have a compulsion to grab it.

True TS.

Exactly like me with cheese n onion bakes when I walk past a greggs.

I had to hide the laser pen... it drove my cat insane. After I'd switched it off he would spend the rest of the evening searching for it.

We have a German Shepherd now and his favourite thing (beside his ball) is the laser pen. He likes to chase it, but he doesn't try to bite it, so I'm not sure what he thinks it is.