This pup sure is one well-trained pooch. Although he's only mentioned in a few lines, he follows the boy throughout the poem, staying by his master's side despite all the shenanigans the hunches put them through.
The pup doesn't get much characterization. But he does belong to a prestigious list of silent, loyal K-9s in Dr. Seuss's works, following in the footsteps of his more famous predecessor Max from How the Grinch Stole Christmas. In fact, the two look like they could be related.
Like Max, the pup sticks with his master through mostly thick and way less thin. Based on his facial expressions in those drawings, we'd have to say he has his own perplexity, disconcertment, and excitement about the Hunches and goes through many of the same emotional states as his master. He even multiplies with the boy and joins in the grand argument at the poem's end to decide the winning hunch.
In the end, like Max, he is rewarded for his loyalty. Whereas Max receives a delicious slab of roast beast, this pup gets to partake in a six hot dog lunch. Not too shabby.