For the best possible 3D audio, you'd want a Mac or Windows machine so that you can install the software. The software will then allow the Moebius to be used to decode and present a 5.1 or 7.1 audio signal (like your AVR does, but instead of sending it to the 5 or 7 speakers, it does its best to create the same effect via headphones.
Outside of that, you can have straightforward stereo or you can have "3D Stereo" which aims to reproduce the effect of listening to a pair of speakers in a room. I've used this with BT directly from the TV in my bedroom and it's actually pretty cool. Not the same as virtual sound, but good, still.
As far as I know, those are the two scenarios - a PC/Laptop for virtual surround, anything else is stereo or "3D Stereo". Happy to be corrected if I am out of date on this.
Might be the impetus you need to build a low cost (but upgradeable) HTPC to replace the Shield....? It'll still run Plex