Is this a rear gear whine?

Maybe my tinnitus is too loud, but the whine is not horrible sounding. Have you tried turning up the radio?

So seriously, does it matter if the car is in N or a different gear? It gets louder with speed, right? Have you checked out the U-Joints, not just for play but for discolored caps or other indications of wear or being frozen? Tires and bearings can also make noises and are worth checking out before settling on a gear problem.

More info is helpful. The stereo fix is not going to work with a baby or wife in the car.
I’d be visiting the shop that put the gears in for diagnostic help. Getting the wheels up and putting it in gear might make diagnosis a lot easier.

I’m still looking for an easier to fix possibility for you that needs checked out first anyways.
The u joints are still a possibly with load on and off noise change, as is the tail-shaft bushing. (You did mention that change in sound in the video.)
I do not know how hard on the gears your driving style is, but I am guessing not that rough with family in it.

At the extremes - a family business work van had missing teeth from the ring gear and did not howl. Grandpa got tired of the fully loaded trucks periodically getting stuck unless he pushed it a bit. But a Cobra I looked at sounded like a civil defense siren at speed from badly installed gears, both with load and when coasting.