This guy wants really badly to believe that animals aren't people, so he's likely biased.
His arguments are:
Recognizing one's reflection and self-awareness are poorly correlated; chimps can be trained to pass theory-of-mind tests but there's no evidence they're using theory of mind to do so. Plus there are humans who definitely are people who still have trouble with self-awareness and theory of mind. So those tests prove nothing.
Animals pay attention to different things from humans, so they don't have humanlike consciousness.
The latter is obviously a confusion between senses of "conscious" as "a valuable mind" and "aware of something". The former is true, but doesn't cover Cambridge's main argument about emotions (states that lead to similar brain activity and behaviors in humans and animals and are described as emotions by humans, and are disrupted in similar ways by hallucinogens).
The Francis Crick Memorial Conference, held in Cambridge last month, has come up with the Cambridge Declaration on Consciousness (PDF).
tl;dr humans still aren't special, consciousness seems to arise in quite a variety of nervous systems and working out what it is is a problem in neurology.
We declare the following: “The absence of a neocortex does not appear to preclude an organism from
experiencing affective states. Convergent evidence indicates that non-human animals have the
neuroanatomical, neurochemical, and neurophysiological substrates of conscious states along with
the capacity to exhibit intentional behaviors. Consequently, the weight of evidence indicates that
humans are not unique in possessing the neurological substrates that generate consciousness. Non-
human animals, including all mammals and birds, and many other creatures, including octopuses, also
possess these neurological substrates.”