The speaker claims to have a twin, and the twin has a dog, but the dog keeps following the speaker home. That small, odd detail is the key. A dog doesn't make mistakes about who it loves, so the animal quietly suggests that the "twin" and the speaker may not be separate people at all.
The last lines turn the joke. "Correspondence" usually means letters passing between two people, but here it also means matching up, like two things that line up. What corresponds is "our two lies." The pun on "lies" is easy to miss: it could mean falsehoods, or the way two things lie side by side. Either way, the twin begins to look like an invented double, a version of the self the speaker made up.
The mood is dry, playful and a little unsettling. The speaker sounds amused, but also slightly evasive, as if they're teasing us about how much of any identity is a story we tell and then keep in touch with.