As human beings, we tend to project human emotions and attitudes onto animals. This concept is called anthropomorphism, and even though it's not scientifically sound, it's still fun to do. In my anthropomorphic opinion, cats embody a laissez-faire attitude, appearing to be unamused and uninterested in the antics of people. Dogs are the opposite. They are spunky and extroverted and want to be all up in everybody's business.
Possums embody something that everyone has within them, no matter their disposition. They're chaotic and filthy. They represent the ID inside our deep animal minds that we can't let out, even in our wildest and craziest moments. Even if you're the most put-together individual who has ever lived, you have a little bit of possum in you. You might be studying for the LSAT, but there's something deep in your brain that wants to roll around in the garbage and bite people. That's why possums are such great subjects for memes. We are possums, and possums are us.