While your household pet could be the vessel for your familiar spirit, it usually is not. Some familiars will possess the physical vessel of plants, trees, or other organic beings, but there are often times when the familiar is not and is held instead within an inanimate object like a doll, a statute or other item. While the familiar has the ability to possess an animal it does not hold a physical body of its own and will blend with the consciousness of the animal it possesses so that it may be held within your pet, but it is not your pet.
Familiars and witches are naturally drawn to each other due to the energy they emit on the astral realm. Once the connection between familiar and Witch is established the familiar will usually reveal its name enabling the energetic bond to occur and allowing each of them to absorb the knowledge and understanding the other has collected over their lifetimes; this will enable each of them to reach a higher level of consciousness.