The puppy episode is significant because it is symbolic of the way Tom views Myrtle as well as how Myrtle views her husband.
Tom clearly does not see Myrtle as his social equal. He thinks of her as beneath him, below him, almost as one as imperious as Tom would think of an animal, like a dog. Perhaps this is why it is so easy for him to strike her so viciously for saying his wife’s name. In fact, he compares Myrtle to a horse when he brags about his sexual conquests to Gatsby. Tom says, “I’ve heard of making a garage out of a stable, […] but I’m the first man who ever made a stable out of a garage.” He is, of course, referencing the fact that Myrtle’s husband, George, owns a garage, and George and Myrtle live in an apartment above the garage. He implies that Myrtle is simply a brood mare for him to have sex with.
Similarly, he behaves in such a superior way with the dog, calling it a “bitch,” as though it is beneath his notice, as he never pays the dog any attention once he has purchased it for Myrtle. Nor does he pay Myrtle much attention outside of their sexual relationship.
Likewise, Myrtle believes her husband, George, to be beneath her as well. She explains that she married him because she “thought he knew something about breeding, but he wasn’t fit to lick [her] shoe.” It’s telling that she uses the word “breeding,” a word often associated with dogs, in her assessment of him. She implies that she knows something about breeding, but the episode with the puppy would indicate that this isn’t necessarily true.
Her friends insist that she was once crazy about her husband. Then, once she has the dog, she basically ignores the pitiful creature. Nick says that “the little dog was sitting on the table looking with blind eyes through the smoke and from time to time groaning faintly.” Myrtle is careless with others, just as Tom is. The puppy seems, in some ways, symbolic of poor George Wilson, as well as of the condescending ways Tom sees Myrtle and Myrtle sees George.