There is no completely solid prof that other species apart from us hunt for sport/fun but examples of this behavior can be found everyday life, for example, cats killing birds or other small animals and not eating them and dogs chasing after things like cat's or even cars, these acts seem to be purely out of sport.

You will never find primary research that says anything is the only anything; because there can be no such research. It's like proving a negative. You will find research showing that particular animals do particular things.

Generally, you will also not find genuine research about human-like motives in animals. You will find research that tries to determine the reasons for various behaviors in animals.

Certainly other species hunt when they are not hungry. They may be doing it for practice, or to learn or teach their young. But it is quite likely that the emotional mechanisms are actually the same as ours. More and more studies make clear the reasonable and obvious theory that our brains work in similar ways and that other animals feel pain, anger, fear, etc. So, also with fun, play, and sport.

Cats are the best known animals that hunt when they are not hungry. Many people think they do it for sport. If you are around cats in a non-urban environment you will probably see them bringing home small birds or rodents that they have killed but have no interest in eating.