Why were there three dogs in the story on hamlet?

There are no three dogs in the story of Hamlet. There is one dog, named "Hamlet", who is a character in the story. This Hamlet is a male dog who is the main character of the story.

The story of Hamlet is not a Shakespearean play. It is a popular children's book called "Hamlet, the Dog" by William Steig, published in 1967. It tells the story of a dog who is given a book about the Shakespearean play and begins to believe he is the tragic hero of the story.

It's possible you are misremembering the story or perhaps thinking of a different story entirely.