I agree, that looks right. My understanding is that these date after the civil war up until 1913 when the military began issuing dog tags. While similar tags were used by some in the civil war they were not commercially made like this one specifically for this purpose.
Added: This particaular design can be found marketed as and used for key tags, even after the ID tag market disappeared.
-does this one maybe say 1st instead of 15th? ...it's a bit hard to read.