Can the spirit of companion animals leaving the body when they transition enter into the owner and become a spirit possession of them, and if so, how frequent is this?

This can happen but happens rarely because the animal typically receives guidance from the higher self and will cooperate with light callers to ascend to the light rather than attach even though there is often a strong desire to be with the human companions. Keep in mind that all deaths are chosen, so at the point an animal exits it has made the decision in conjunction with the higher self to leave, so it is not the same as a human who is highly distorted many times at the time of transition and is unable to cooperate or successfully complete the process. This happens much less frequently with animals, so this would be a rarity.