Though "Anecdote of the Jar" focuses on Tennessee, Tennessee is, after all, one of the United States. (At least, the last time we brought up Google Maps it was.) The mention of the name of a particular state, we're guessing, has more to do with American and human ideas of the natural world and human borders than a specific attribute of the state itself. Though this poem has a lot to say about society in general, it specifically takes place in the country of pilgrims and pioneers who were, like the jar, conquerors of an untamed land—America.
Bush vs. Jar! The poem shows that the conflicts between the man-made and the wilderness that may not be unique, but is representative of America at large.
It's no accident, gang. The inclusion of the name of a state involves American culture into the ideas in this poem.