What kind of photography is this?

A Daguerreotype would be an image viewable by differential reflection on a silvered copper plate; it would surely not resemble film pasted on cardboard.

The upper image is most likely a print. If actually made in 1880, it might have been from a collodion negative, or a dry plate, or a Kodak paper negative (from the first consumer camera) or just possibly then-new celluloid film. Most silver image prints of the day were albumen process, and tended to a very warm tone, but gelatin was just getting started (originated in the early 1870s) for printing, as well as plates and film, so it might well be a gelatin print -- and as suggested quite possibly a photograph of a professionally drawn portrait (from the days when that was actually cheaper than a photograph, or before 1840 when there were no photographs yet).