The Shield is also one of the only Android TV devices that passes on HD audio codecs, so if you stream any Blu-ray rips with DTS-HD Master Audio or Dolby TrueHD and you want passthrough to work correctly, you probably want a Shield.
A shield is just a client, can't be used as a server AFAIK.
DTS is just a different set of audio formats/standards, originally popularized in theaters by the movie Jurassic Park.
DTS-HD Master Audio is their lossless audio format, meaning it decompresses out to being the exact same audio bit-for-bit as the mixer heard when they finalized the audio for the movie.
Likewise, Dolby TrueHD is also a lossless digital audio format, so it also decompresses to the exact same audio the mixer heard when mixing/mastering the movie.