How does the narrator acquire the dog named Don in Chapter 19 and what is the significance of her acquisition?

When the narrator's neighbors on one side are taken away by the Memory Police, the officers somehow neglect to collect the captives' dog, as they usually would when carrying out a raid. So the narrator decides that she will adopt the dog whose name is Don, an act she knows is somewhat dangerous because the Memory Police have records of everything, including the existence of pets belonging to captured parties.