In The Stranger, the relationship between Salamano and his dog is analogous to both the relationships between Mersault and his mother and between Raymond and his Arab girlfriend. The dog reminds Meursault of his mother; therefore, the dog is a symbol for that which is lost, dead, or abused.
First, Salamano is more heat-broken about losing his mangy dog than Meursault is about his mother dying. This is not to say that Meursault does not love his mother; on the contrary, it shows that he appreciates his mother's life and refuses to excessively mourn her death (the way that Salamano weeps over his dog).
If The Stranger is an existential indictment of people who choose death over life, then Salamano is guilty as charged. He seems to care more about the dog now that it is gone than he did when it was alive. Camus is morbidly poking fun at the fact that most people, like Salamano, are addicted to the culture of death and guilt. As a result, they refuse to live meaningful lives.
Secondly, the relationship shows how violence can damage relationships. Salamano would beat his dog simply because it is old, just as Raymond would beat his girlfriend simply because she is a woman. Both men harbor violent tendencies toward weaker creatures, most likely because they are addicted to the culture of death, guilt, and suffering.
On the other hand, the absurd hero Meursault chooses the simple pleasures of life: he enjoys the sun, the beach, smoking, and sex; as a result, he does not need to engage in a power struggle with those who are defenseless.