Yes, according to Hindu mythology, Shiva does have a pet. Shiva is often depicted with numerous animals and birds as his companions or vehicles, known as vahanas. One of his vahanas is a dog.
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In his aspect as Bhairava, an intense and fierce form of Shiva, he is often portrayed riding a dog. This is mentioned in the Mahabharata, one of the two major Sanskrit epics of ancient India. The dog serves as his faithful companion and carries him during his travels.
Another deity associated with a dog is Khandoba, who is considered an avatar of Shiva in some regions of India. Khandoba is a warrior god and is often depicted riding on a dog, which is also his vahana.
Both these depictions of Shiva and Khandoba with a dog as their vahana highlight the significance of dogs in Hindu mythology. Dogs are seen as loyal and protective animals, symbolizing vigilance and faithfulness. By incorporating a dog as a vahana, Shiva and Khandoba exemplify these qualities.
Overall, while Shiva is primarily associated with Nandi, the bull, as his main vahana, the depiction of him with a dog as his vehicle adds diversity and symbolism to his mythological representations.