It is true that domestic cats are related to lions, but only distantly. In fact lions are more closely related to tigers than your average house cat. Domestic cats and lions and other bigger cat species are in different subfamily, but still fall under the family Felidae. For instance, domestic cats are in the subfamily or genus Felinae, whereas big cats such as the lion, tiger, leopard, and/or jaguar fall under the subfamily Pantherinae. The cheetah on the other hand belongs to the genus Acinonyx and is the only living species.
All cats, however share the same types of instincts, though some animals in the cat family are much larger than others. Domestic house cats have the type of diet and digestive system as their wild counterparts, however most people choose to feed their cats hard or soft cat food, which is technically species inappropriate. Most foods contain too little meat and/or too many plants to truly benefit a cat in the way that a meal should. Some outdoor cats do hunt their own prey (mice, birds, other small rodents) in the same ways as the lion or other members of the cat family. Cats get their liquids from the foods they eat.