I think we need to sustain and not to collapse the tension between these simultaneous truths -- "we understand each other perfectly" and "we don't understand each other at all" -- if we want to flourish together. More, I think we should celebrate it. In the history of our relations with other animals, and particularly in the history of our domestication of other animals (and their domestication of us!), views have tended to swing from one pole to the other, from the conviction that other animals exist only as extensions of human need (or of human fear, as in the case of the benighted wolf) to the conviction that they exist utterly apart from us. Wittgenstein's oft-quoted aphorism captures the latter belief nicely: "If a lion could speak, we couldn't understand him." Likewise, Thomas Nagel's famous (and to his mind impossible) question -- what is it like to be a bat? -- encourages an all-or-nothing judgment on the possibility of shared experience. But otherness is always radical, and subjective feelings of connection are always an objective illusion. (Or rather, the connection itself is illusory, though the feeling would probably show up on a brain scan.) We have no direct means of access to any other being's perception of the world, no matter the species, so unless we wish to retreat into lonely solipsism, we have to make do with indirect means and earnest approximations.
With that limitation in mind, we have good reason (founded on objective evidence) to suppose that, in many ways, our pets' and other animals' emotional and cognitive experience strongly resembles our own. The speaker whose talk I'm most excited to hear at the upcoming Clicker Expo is Jaak Panksepp, the neuroscientist who holds the Baily Endowed Chair of Animal Well-Being Science at Washington State University (wonderful that there should be a chair so endowed). He is one of a small (but happily increasing) number of scientists who dare to emphasize the obvious homologies (common structures with common origins) among diverse animal minds, especially among the minds of humans and other social mammals. (I use "minds" advisedly, as Panksepp is interested in subjective as well as scientific modes of inquiry and description.) He is also leading research into homologies that are not so obvious, teasing out the physiology and chemistry that underlie those brain processes that we hold in common (and variations that we do not). His book Affective Neuroscience is a marvel. Densely technical in places, it nonetheless serves both as an excellent overview of contemporary research into the dynamics of primary emotions (including their influence on cognition and learning) and as an eloquent, richly speculative description of the big questions that remain.