I'm pretty sure the game you describe exists in the US, but being a sweet little child
, I never did such a thing so I don't know the name.
As to cats--Heinlein wrote a fun novel in the 50s, called The Door Into Summer
which opens with the human narrator talking about the place where he and his cat live (old farmhouse in Vermont if I remember correctly). The problem is the house has 11 doors, and the cat insists on checking each door (in the case of bad weather) looking for the door into summer. Every cat I've ever had seems to have this idea--somewhere there's a door into summer.
Then there's the in & out game, which is the cat version of your knock and run game. Paul Gallico describes it in The Silent Miaow
which is out of print I'm afraid. But th book purports to be a training manual for kittens, written by a cat. The in & Out game is designed to keep the human coming to the door all the time.
Still, I wouldn't trade Emily and Victoria for anything (except Tony).