I'm guessing an iron-on transfer...possibly on cavas. They do this at Kinkos.
I used to work at Kinko's and I was actually the one who printed on canvas, watercolor paper, backlighting material (cool stuff), tyvex, etc... and these animals are not canvas. It's super stiff, and brittle, and the ink kinda cracks if you roll it too tightly, so sewing and turning it inside out is out of the question. (Though I sure tried...ohh...miss that job...) (AND I think it's $18 a sq foot.)
I personally think these were silkscreened, because it seems more professional than inkjet transfers AND
you can use any color ink
make multiple prints
make any size
get a better fabric for sewing (not stiff, better drape, etc)
get a richer print
can print on fabric that may melt
Anyway, that is how I think THEY did it, though I guess a transfer would be easier.
when I was a kid I tried setting up a silkscreen exposing unit (It's a lightsensitive process) with tinfoil and pieplates and a garden hose - and had no luck, but I eventually took a class and it was real easy with their fancy setup.
Maybe I'll try the old-skool way again... I'm digging these pillows.