My husband's ancestor Johannes (John) Hesse was born in 1842 (father Johannes Hesse and mother Elisabeth Pfuhl). His father died in 1849. He had an older brother Heinrich who stayed in Germany. So I think Carl/Karl/Charles was the youngest son so possibly born 1843-1850.
The original of the photo is held by the German branch of the family. On the reverse is the pencilled name Chas Hesse. The photographer was E. Thibodeau. The Minnesota Historical Society (MHS) has a Directory of Minnesota Photographers online. The entry for Edward Thibodeau shows him as a photographer in Minnesota from the 1880s through to the 1920s. For the period 1890-1894 he was operating from 28 Central Ave Minneapolis. Reading up on Nicollet Island suggests that this address is a later version of the address on my photo, 28 Nicollet Island. Therefore I suspect the photo was taken in the 1880s.
Going with a birth date 1843-1850 would make the man in the photo about mid 30s.
I only assume he was in Minnesota because he had his photo taken there. I checked the Minnesota Census Index on the MHS site but could not pick any likely candidates. One name did leap out - Ferdinand Hesse. John who came to Australia named one of his sons Ferdinand - never could understand where that came from but maybe there is a connection somewhere.
I was hoping that identifying the uniform might give me some clue about where to search next.
Sorry for rambling on - I'll post to military as you suggested.