Cats love affection, but particularly they love being rubbed on their cheek. I guess they have scent glands there which give them pleasure when rubbed. That would be an evolutionary mechanism to make them want to spread their scent around.

So I propose a machine made of two soft rollers, upright facing each other, with a pressure sensitive pad in the middle. The cat presses the pad and the rollers rotate, one pressing against each of the cats cheeks. Think of the two upright rollers of a very small car-wash to visualise the two rollers.