The idea that a pet is the same as a familiar came from the centuries when poor women who had only a pet for company were often accused of being Witches.. It has continued in modern times because of the crop of poorly researched books on the market these days. But the fact of the matter is that our beloved pets and a Witch's familiar are very, very different things.
A Witch's familiar is an animal spirit or a magical construct created by the Witch or the Magician as an adjunct to their magical workings, as a repository for power, and if needed as a defense against magical attack from another magic user. A Witch or a magician can draw upon the energies of the familiar when they need an extra boost of energy in a magical working. Conceivably if the spell is important enough, the Witch could completely drain the familiar of energy causing it to "die" or cease to exist. And in the case of a magical attack, the attack could be diverted to the familiar so that it would be the familiar which is destroyed rather than the magic user. None of these are things that I think any of us would want to do with our pets.
There are excellent articles on Familiars here:
Familiar Spirit - http://en.wikipedia.org/wiki/Familiar_spirit
On Fetches and Familiar s - https://www.spellsofmagic.com/read_post.html?post=857694
Animals are very sensitive to the energies around them and they may be drawn to us when we are working magic because they can sense the energy. They also want to be around their people, which again will make them want to be where we are..and that may be where we are doing magic. Neither of these things makes the animal a Familiar or inherently magical in and of itself.