This is by far a difficult question, since most of the pets that are the subject of photography are almost every time trained to sit and offer particular poses. Therefore, in order to answer that question, pet photographers need to refer to photoshoots that implied special props and difficult positions for the animals involved, in which case the best pets to be photographed are the ones that are the most enduring when striking a particular pose for a scene.
Having said that, the pets that are static when they are at their most usual atmosphere could be the called as the easiest to be photographed. One pet in particular comes to mind and that is the iguana, which by all means can be seen as a calm and enduring animal. Not only does it maintain a position for an extended period of time, but it even rarely blinks, making it the idea subject when dealing with static photography. Modeling the posture of iguanas is also an easy task, thanks to their inherent flexibility and the ability to maintain a position even in upside-down positions, when hanging off branches or other types of props.
Cats, dogs and other fluffy animals such as rabbits or Guinea pigs all make for wonderful photography subjects, just that almost certainly they need to be “lured” into assuming a specific posture and constantly motivated in order to maintain that position. Most of the time, food and special attention are required in order to keep the pets in the required position just enough time in order to take the photographs.
Provided that the photographers have the necessary skill and accessories needed in order to make the pets stand still and maintain a posture for long enough to capture an image, basically any animal can be the ideal photography subject.