I found the video that first brought it to my attention
At 0:45 the narrator mentions how the diver is covering the regulator when switching and after training at 3:40 he mentions how she's now holding the regulator "correctly". I've searched for videos on replacing the regulator and have found that instructors that show how to do it in neutral buoyancy tend to used the hose grip and those that demonstrate from a kneeling position tend to hold the reg itself. That's what got me wondering if it was agency or tech diving related.
You're probably right in that it's to reinforce correct technique when sharing the octopus, I should probably have thought of that