You know how a square can be a rectangle, but by definition a rectangle can't be a square? Well Oppel chooses to draw parallels between Matt and the cloud cat in a similar way.
The cloud cat is a creature that was never meant to dwell on land—its species normally does everything in the air: eat, sleep, play, and even give birth while aloft. Unfortunately, in the beginning of the book we witness our cloud cat being born with a birth defect that renders her incapable of flight. When Matt and Kate meet her she has adapted to living amongst the trees but it will never be the place where she was meant to be.
Matt would probably say that he can totally relate to the cloud cat. He feels like he belongs in the air—he was born while aloft, and he plans on spending his life that way too. So if you're paying attention, Oppel is using the cloud cat as a characterization of Matt's inner self. Matt feels like he is a wild creature trapped on earth, just scraping together an existence until he can (re-)learn how to fly.
The problem is, despite his discomfort, human beings are designed to live on the ground. His urgent need to be airborne was created as a way to cope with his inability to mourn the loss of his father, not from some biological imperative.
The great thing about this extended metaphor is that in the end everyone ends up where they should be. During his battle with Szpirglas Matt realizes that he can't actually fly and is filled with disappointment in himself:
I could not fly. I had crashed. I was not lighter than air after all. I'd fallen, and a great shame seeped through me. (20.56)
Then over time he becomes aware that everything is okay:
But then, unexpectedly, one night I'd dreamed of my father, even though I was landlocked. I was flying alongside the Aurora, and he'd come and joined me, and when I woke up that morning, everything was different. As long as I could still dream about him, I knew everything would be all right. I didn't need to be aloft to find happiness. (21.52)
But it isn't merely coincidence that at the exact moment he learns that he cannot really fly, our cloud cat is freed from her fear of flight and takes off into the sunset with a flock of her own kind. It is symbolic of Matt's delusions being released so that he can finally learn how to find happiness within himself.