Symbolism, Imagery, Allegory
Let's have a look at how the dogs in the story, Medji and Fidele, spend their time. They sit within the confines of their homes (8.6), are pampered, write each other elaborate letters about their favorite dishes (8.8), talk trash about the lower classes (8.8), discuss but don't understand the significance of all those political affairs the men are involved in (8.12), and gossip about their suitors (8.18). Does that remind you of anything? If you were about to say "upper class ladies," but hesitated, we'll just put it out there: yes, the letter-exchanging dogs do indeed symbolize upper class Russian women.
If you're surprised by the misogyny (fancy word for woman-hating) of this bit of symbolism, it's not like Gogol is otherwise kind to women in this story. Look at this: "Oh, she's a perfidious being—woman! Only now have I grasped what woman is. Till now no one has found out who she's in love with: I'm the first to discover it. Woman is in love with the devil" (13.1).
And if you're wondering how accurate this "doggie" portrayal of the lives of upper class Russian women actually is, we'll say Gogol nails this one. Just replace the "doggie" bits with their human equivalents. It's especially on-target, for example, when it gets to the kinds of things men were saying about women's writing.
Women had started to have a more active role in Russia around the time this story was written, and were getting criticism like Poprishchin's from men: "Never in my life have I heard of a [woman] being able to write. Only a gentleman can write correctly" (1.2), "Such nonsense! As if there were no better subjects to write about" (8.9), "Extremely uneven style. Shows at once it wasn't written by a man" (8.17), "Pah, devil take it! … What rot! […] Give me a man! I want to see a man; I demand food—such as nourishes and delights my soul; and instead I get these trifles" (8.19).