The witnesses on the typed version (not the diagram) might give us a clue - one is "W. V. BRINGLE". A Walter V. Bringle was married in Tipton County, Tennessee, in 1896; I can;t find him in the 1900 census but by 1910 he was living in Covington, Tipton County, and practising as a lawyer.
It turns out there was a Munch post office (1880-1907), also in Tipton County (Tennessee Place Names & Post Offices). The USGS have an entry for it which places it somewhere in the vicinity of Gilt Edge, a few miles west of Covington. It is not identifiable on this 1888 map of Tennessee post offices; when postal service was discontinued, it was served by the Brighton post office (SW of Covington)
Tipton County is quite a way from Jackson County, but the similarity of "Munch" to "Munich" is intriguing, as is the presence of a local lawyer with a very distinctive name - exactly the sort of person you might expect to draw up and witness a patent application. (Of course, you might also expect him to get his own county right, so take this with all due caution.)
I can't work out exactly where Munch was, but the postmaster in 1895 had another distinctive name - "Thornton G. Ladd". We can find him in the 1880 and 1900 censuses, living in "civil district 2" of Tipton. Not a large area - there are a bit under 800 people. And in the same civil district in 1900, so presumably in the immediate vicinity of Munch, we find...
Andrew Jackson, b. Jan 1874 - railroad labourer, can read and write, owns own house, married with one small daughter.
By 1897 the postmaster had become John C. Gracey; in the 1900 census he is resident in "civil district 8", which seems to be somewhere in the vicinity of Covington; however, the only Andrew Jackson there (b. May 1832) is illiterate and so seems unlikely to have been the patent author. There are only a couple of Andrew Jacksons in the county, so it might be plausible he is the other one's father, but that's very speculative.
I don't think it's cast-iron evidence, but it does seem to match the details given - "Munich" is an easy slip for "Munch" (and, as Mark notes in comments, it is easy to see how "Munch"/"Munich" could result from confusion about pronouncing Munich/München). While getting the county wrong is a bit strange, it is at least sort of explicable that they could just have written the guy's name down twice.