I've got friends who refer to this sort of thing as 'half-harmony'--harmony half the time, same notes half the time, more or less. Seems to be fairly common in country-type music here; even duos such as the Louvin Brothers who were known for their harmony do it some of the time. I don't recall if/how often it appears in shape-note, though. It's been awhile since I did that.

(On the other hand, we do not consider the same note an octave apart 'harmony'. I don't know what the official 'rules', though.)