Although there as several omens of dogs and death (a howling dog predicts the death of a loved one) there are no omens, I can find, about the death of a dog meaning anything.
The breed of dog in The Omen was a Rottweiler.
It means someone or something indicates impending death/destruction. Think of it as an omen.
An omen
Belgian shepherd
The family dog is a Rottweiler and the dogs in the cemetery were Dobermans
death i think
banshee
"When dogs cry at night, ghost is around the corner." When a dog howls in an otherwise silent night, it is said to be an omen of death, or at least of misfortune. A howling dog outside the house of a sick person was once thought to be an omen that they would die, especially if the dog was driven away and returned to howl again. A dog which gives a single howl, or three howls, and then falls silent is said to be marking a death that has just occurred nearby.
The hawk was an omen of death in ancient Greece.
an omen or prophecy
good or bad
No, an omen is a sign, which can mean good, bad, dangerous, etc. And a warning is something that is going to happen. For example the clouds are dark grey, that is a warning that a storm is coming. A omen is something religious and can be and mean different things to different people.