I found this on the website Straightbourbon.com
As with French-appellation wines, there are strict laws governing just what a Bourbon must be to be labeled as such. For example, at least 51 percent of the grain used in making the whiskey must be corn (most distillers use 65 to 75 percent corn). Bourbon must be aged for a minimum of two years in new, white oak barrels that have been charred. Nothing can be added at bottling to enhance flavor, add sweetness or alter color. Though technically Bourbon can be made anywhere, Kentucky is the only state allowed to put its name on the bottle. And as Kentucky distillers are quick to point out, Bourbon is not Bourbon unless the label says so.
I'm not sure how Jack Daniels is made, but if it meets the qualifications for bourbon then you are right about the use of the state name. Hope this helps.