The driveshaft normally has some play. On a 2003 evo, the driveshaft rubber bushings are probably a bit worn as well. That would not cause the issue in the video.

 

It's the transfer case

 

Transfer case. Make sure your mechanic filled the transfer-case properly. Even though he drained and filled the T-case w/ new fluid, doesnt mean he did it right... Mechanics unfamiliar with Evo specific maintenance procedures, usually dont know filling through the "fill" hole just doesnt cut it for the Evo transfer case. (unless you wait 20 min/ refill or drive around the block/ refill..) 90 wt. gear oil takes forever to make its way through pencil sized holes/ passageways.

Everything from this thread/ video is pointing towards your T-case having the incorrect amount of fluid IMO.. More than half the people who change their fluid, dont fill it up properly or measure the amount of fluid going in/out.

Correct filling procedures..

http://www.jackstransmissions.com/pages/t-case-filling

 

Tcase, as said above check the correct fluids were used. Another easy way to tell is to check the ring and pinion Play in the Tcase, mine had about a 1/4 inch ! and the whine was loud and at all mphs

get the car up on jack stands all 4 wheels up, rotate the front wheel back and forth hear and feel that play?? thats the Tcase ring and pinion saying "replace me with a hardend unit"

 

I would guess t-case. I had mine go a month after I got the car. If the wrong lubricant was put it it at one time that didn't have the right heat specifications it could cause that problem.

 

Nice I would jsut get it rebuilt or have it inspected by a real professional and stop using different kinds of fluids and get with the diaqueen if you can or an oil that holds the same weight as it atleast. If you never had it done now is the time. Send it to Jacks not Shep I speak from experience.

 

Sound like my evo x at 15 k miles
But my noise is deff from trunk area so im pretty sure its my rear diff

 

Get the filler tool from jacks they are great and fill it up with the link thats on their website that bekevo put up.

 

I'm curious if there's a universally accepted t-case failure diagnosis method.
EG: drive at X speed with 1/4 throttle input, if noisy, replace, or something similar.
There always seems to be a disagreement, some say "normal been that way since 03" and some say "it's time to rebuild"
Regarding the OP's video, that seems to be a safe bet it's on its way out. But a few posters described a slight whine @ 55-60.. is that normal/acceptable/terrible/imminent failure?

 

There seems to be a 10mph window of normal slight audible feedback from the drivetrain from 55 to 65 mph depending on tire size. If the normal range of audible feedback begins to widen, then it is time to prepare for a rebuild.