Cats which are brought up in a loving household are often faithful to their household or to particular individuals. Because this has to be earned by the owner, it is a far more precious thing than a dog's submissiveness. Cats also look after themselves and will visit neighbours to take advantage of company, food, warmth etc, but most will return to their own household if you are providing a comparable level of care. This is because a cat is a solitary hunter and has evolved to take advantage of a food supply it can exploit alone. Dogs hunt as packs and the pet dog sees you as a companion hunter - its survival prospects are best served by staying with you, however you treat it.
Cats often bring home their hunting trophies as a sign of appreciation.
It is their contribution to a food supply - just as they would provide food for their kittens or for communally reared kittens. They also consider your home to be a safe den where food can be eaten without the threat of being stolen. The fact that you do not appreciate the gift, or you release it is possibly viewed as you being a rather wayward and inept kitten though many owners give their cat some cat food to compensate for the lost prey item. Cats can be trained, but unlike dogs they demand tangible rewards such as food, not just lavish praise. A cat's behaviour is not set up so that praise alone is a sufficient reward.