Post-ACW - through the 1890s, rifle clubs existed in profusion - both on these shores and overseas, as well.
The uniformity of the medals suggest they're from the same club/district/region, as they look like the same maker produced them.
I doubt there's a truly military connotation - more like one of the 'society' military organizations like the 'Society of the Army of the Tenessee', as an example - it isn't 'Grand Army of the Republic' - that uniform was laid down in their Regulations.
Vaya,
Scouts Out!