The Shinx line, being based on Lions, are meant to have somewhat of a Lion biology. Luxio live in groups in grassy areas, they hunt for food (meaning they must be carnivorous and eat other wild Pokemon), and are able to communicate with the group through electrical pulses from their paws.
One may envision that Shinx are the lion cubs and have not yet matured to fully-grown Luxray. Luxio may be the adolescents of the group, aiding in taking care of the cubs or possibly being the ones that do the hunting. We do not have a lot of insight into how these creatures may interact in their supposed natural habitat, so it is mostly speculation.
Lions live in what are known as Prides. Usually in a pride, there is one older male and several females and cubs. When male cubs reach a certain age, they fight to take over the pride or they leave and start their own, so very rarely are there multiple adult males in a lion pride. We can assume that a Luxray pride may work similarly. Perhaps there is one Luxray (Adult) that oversees the pride and possibly many Shinx and Luxio. When a Luxio evolves, they may have to fight for the pride or leave as well.
As to what Luxio hunts, that is another mystery. Lions are hunters and scavengers and will eat most things in the savannah. They will hunt types of gazelle and steal kills from other predators, they may even find Cheetah cubs and kill them. They could not be seen as merciful animals. Would Luxio hunt other Pokemon to devour? In other areas where you find Shinx or Luxio, other Pokemon you find would be ones like: Buizel, Bidoof, Abra, Starly, Shellos, and possibly Kricketot. It seems like Buizel and Bidoof might be ideal meals of choice for a predator like Luxio. Large cats tend to go after smaller mammals and less after birds or insects.
Though Lions may have been the design inspiration for Luxio, we have to remember that the world of Pokemon, even though it resembles Earth, is very different in all its versions. The anime version is fairly different from the game version. In the anime, these creatures are sentient and only seems to eat berries and people's food. They can communicate with people verbally and seem to understand things the way a human would. In the game, they seem to be more like pets. In the end, they could be very distant from their earthly counterparts.