The word is fairly generic, and seems like it could easily have been applied separately to a number of different things. For example, I found this from 1969 — "A new electronic method called Chromography re-creates paintings in such exact detail that experts find it difficult to tell the Chromograph from the original." — but I think this booklet from 1871 is probably most relevant. That certainly fits your Civil War era identification. From the booklet:

PHOTO-CHROMOGRAPHY

By this simple process any person unaccustomed to painting, and ignorant of art, may colour photographs, and produce, with rapidity and little trouble, effective, permanent, and beautiful pictures, so soft and delicate as to closely resemble painting on enamel; may render the treasured family portrait doubly valuable by addingt he warm tints of life to the faithful but cold and deathlike production of the photographer, and produce a pleasing as well as a truthful representation. The largest and the smallest work maybe painted with equal facility — the life-size portrait, or a miniature for a locket — the only qualification for success, even in very elaborate pictures, being taste in the arrangement of the colours.

The booklet consists of half a dozen pages of instructions, and then as many pages of advertisement for materials.

The process described is to apply a "Diaphanous Varnish" to the printed photograph so the paper becomes translucent, and then paint on the back side. It is not at all as automatic as the flowery description indicates. (For example, the instructions say "First apply the appropriate colour (pink) to the lips and cheeks very sparingly, softening the latter with a dry brush.")

I expect from the quote on your advertisement that some enterprising individual took this and decided to make a business of it. However, the process described in this booklet is destructive, and were it applied to the photo sent in, returning the original would have been impossible. Perhaps the entrepreneur first made a photographic reproduction, or perhaps they're using the same word to describe an entirely different process of coloring.