As noted, the NSX drivetrain does appear to have a lot of gear lash in the drive train. After I got my 2000 in 2011, I did a similar rear end up in the air 'spin test' because I thought the 1/2 shafts must be toast because there seemed to be a lot of jerkiness in the drivetrain. Doing the test confirmed that the lash was not originating with the 1/2 shaft tripod joints; but, in the differential / transmission. The local dealer still had an NSX certified tech in 2011 and I quizzed him about this when having some maintenance done and his response was 'its normal - they all do it'. I think the problem is compounded by the fact that the NSX has a very high (and narrow) clutch release / engagement which compounds the lurching and banging that can occur in a first gear roll off. This lash is really noticeable if you try and hold a low speed (10 - 20 km/hr) in first gear. Let off the throttle and then apply a little throttle and you get this click - click jerky driving motion. It is the reason I hate driving the NSX in stop and go traffic. I have observed that some owners deal with this problem by revving the engine and slipping the clutch which reduces the jerking; but, shortens clutch life. Smooth take -offs from first gear in an NSX is a bit of an acquired skill.
As noted by RYU, what does appear abnormal in your car is the amount of noise associated with the drivetrain lash. In my car it is more of a slight click as you rock the wheel back and forth where as your car 'appears' to be much louder. However, that could be an artifact of using a cellular phone to capture the audio. From the video clip, I take it you are doing the spin test with both rear wheels in the air and the transmission in neutral? I suggest that you repeat the test with only one wheel in the air and the transmission in gear so that neither the transmission input shaft or the other wheel are free to turn. Then, by rocking the raised wheel back and forth you can get a sense of how much rotational lash is originating between the input to the gear box and the output of the differential. You can repeat this test using different gears to determine if the amount of lash changes between gears which might be a sign of excessive wear in one gear pair. It has been 13 years since I did this on my car so I can't really remember how much movement I had. The car is in storage right now so I can't check; but, some other member might be able to advise as to how much they have. You can then assess whether your movement is excessive. When you do this test, if you can have somebody slide under the car and hold their hands on the outside of the inner and outer tripod joint housings while you do the rock back and forth test. They should be able to detect relative movement in the tripod joints which would be a sign of tripod joint wear.
Depending on how much total lash you have this may be normal drivetrain lash exacerbated by some some wear as RYU suggests. Measure the actual rotational lash at the rear wheel with the other wheel captured and the transmission in gear. If you can, compare to lash measurements from other owners. If the lash is not excessive and your drained transmission oil comes out clean then maybe this is just normal and the nasty noise is an artifact of the cellular phone recording. If the lash appears excessive then it may be necessary to start removing things for close inspection. If the oil has been in the transmission for a while I would be strongly guided by what the oil looks like when it comes out of the transaxle case.
Are you using Honda manual transmission oil in the transaxle? If not, and by chance you are using something like straight up Red Line MTL that can cause some new noises to originate from the differential because it tends to make the diff behave more like a closed differential.
