Has your dog had a bark-mitzvah yet? Just kidding, but seriously there are very real aspects of Judaism that do apply to a pet of a Jew. For example Shabbat and eating kosher. Let me explain.
The Torah states: "The seventh day is a Sabbath for the Lord your God, you shall not do any creative work, you, your son and daughter… your animals, and the stranger that is in your midst."
This means that an animal is not allowed to transgress one of the laws of Shabbos, just as a person must not. Obviously, animals are not obligated to monitor their own Shabbos observance, but their master is. For example, he may not allow his animal to carry something in a public area. An animal is allowed to walk in a public area with protective clothing or a leash which has a protective purpose, just as a person is allowed to walk outside while wearing clothes. However, something that is an extra burden (e.g., a dog ID tag) is forbidden.
Keeping kosher. Another issue of concern for the pet owner is pet food. A Jew is forbidden to derive any benefit from either a mixture of milk and meat throughout the year, or from chametz on Pesacht contain these ingredients.
As long as we are on the topic of pets and pet food, I thought to mention Rabbi Pinchas ben Yair's donkey. According to the Talmud, his donkey would never eat untithed grain. It could sense that the food was inappropriate and would decline of its own accord, without being told that it was forbidden. The Jerusalem Talmud tells us that this is an indication of a righteous person - that not only will he never accidentally eat something that is not proper but even his animal will not.
