Yes, it is. Dogs may not understand the story in the same way as humans, but they can respond to the tone of your voice and the emotions you convey while telling the story.
Perhaps in the 'dog star story', there was a tribe that worshipped the stars. They especially revered the Dog Star. One day, a strange dog appeared in their village. The dog had markings on its fur that resembled constellations. The tribe believed that the dog was a manifestation of the Dog Star. The dog stayed with the tribe and helped them in many ways. It warned them of approaching enemies by barking at the appearance of a certain starry pattern in the sky related to the Dog Star. The dog became a beloved and important part of the tribe's story and legend.
One possible plot could be that the dog accidentally ruins the boy's important project. For example, the boy has been building a sandcastle at the beach, and the dog runs through it, screwing up all his hard work.
One possible ending could be that the dog's barking attracts a group of people who then pull the man out of the well safely.
One possible theme could be friendship. The little dog might make friends with other animals or even people in the story.
A girl, let's say, Rose, saw a dog at the beach. The dog was chasing seagulls. Rose joined in the fun. The dog was so energetic and seemed to like Rose immediately. They played fetch with a stick in the sand. After that day, Rose made sure to visit the beach often to play with that dog. It was a simple yet lovely encounter between a girl and a dog.
Yes, it can be useful. Dogs may not understand the story in the same way humans do, but they can sense the tone and emotions in your voice. A calm and gentle voice while telling a story can have a soothing effect on them.
There was a kid called Lily. Her dog was a cute little puppy named Charlie. Charlie was always getting into mischief. One time, Charlie found Lily's schoolbag and pulled out all her books. Lily was a bit mad at first, but then Charlie started wagging his tail and looking so adorable that Lily couldn't stay angry. They ended up playing a game with the books, stacking them up and knocking them down.
Once there was a family dog named Max. He was a golden retriever. Every morning, he would wake up the kids by licking their faces gently. He loved to play fetch in the yard. One day, a little rabbit got into the yard and Max chased it around but didn't harm it. Instead, he just herded it towards the fence so it could escape. Max was truly a beloved member of the family.
There was a dog named Buddy. His first adventure was when he went to the park for the first time. He saw so many other dogs and people. He was really excited and ran around sniffing everything. He found a stick and brought it to his owner, looking for a game of fetch. It was a great day for Buddy and it was just the start of many more fun outings.
It might be that a neighbor has a dog that the main character sees every day. One day, the main character had a particularly nice moment with the dog, like sharing a snack or playing a little game. Then, later when they saw the owner, they wanted to convey their continued positive feelings towards the dog through the 'tell your dog I said hi' message.