Is this the proper application for a drive through an extension box?
or is something missing????????

usually you use the inner transom plate for its integrity of all the bolts fastened equally in the same plane. this also holds the drive better and is less stressful on the transom.
what is in the box? is it a merc carrier or something homemade?

If they used a offshore style rear mount, then the ear mounts wouldn't be needed. I've seen the stock plates distort significantly from being over tightened, so his setup doesn't concern me. The boat was obviously rigged for a box as they hole isn't a merc spec one.