The Dagor 467xxx serial number places its year of manufacture about 1921, about five years before the forcible assimilation of Goerz into the post-WWI Zeiss-Ikon conglomerate and at least 25 years before introduction of the Prontor SV MX-synched shutter ( the X-sync contact is also visible in the OP photos). So, as others have noted it's definitely an older lens remounted into a more modern shutter.
As to coverage. a 16 CM Dagor, aka 160mm +/-, will have a sharp coverage angle of at least 75 degrees at normal-taking small apertures such as f/22, likely an even higher coverage angle at f/32 and f/45. That 75 degree angle translates into an coverage circle of at least 240mm, more than enough to coverage 5x7 with some movements. An actual 210mm Dagor would have even more coverage, well beyond 300mm image circle.
As to why B+J listed this or a similar lens in 1938 as a 4x5 lens, one can only surmise that this was because 160mm is a close approximation of the diagonal of a 4x5 negative and hence within what would be considered a normal lens for 4x5 use, albeit with a lot of coverage.
As others have noted, older Dagor and Protar VIIa lenses in good condition are frequently eminently usable today. If you're uncertain about that, just look at same of the technically excellent images produced by Ansel Adams and Edward Weston in the 1930s and 1940s with such lenses. I've seem very big enlargements of Adams images taken with Dagors and Protar VIIa lenses and could not find fault with the resolution or image contrast.
