I like it! (I confess it took me a second. "Emby? Why Emby? OHHH". Don't tell anybody). It's short, it's whimsical, the GF thinks it's cute. Rolls off the tongue better than "MEDIA BROWSER" for her to say "I want to watch Hot Tub Time Machine, is it on Emby?" More friendly and anthropomorphic.
Kodi, yeah, well, not a huge fan, but XBMC is hard to type and impossible to say, so definitely an improvement. (Although is it Low-dye, like Lodi, California? Or Low-dee? Meh). We usually say "in the TV room" (Kodi) or everywhere else (MB/Emby).
Most importantly it's unique- googling for "media browser" or "mediabrowser" you end up with a panoply of random unrelatedness, Kodi and Emby don't suffer from that. The domain is always tough; yeah, it's pointless, except people are JUST getting used to .tv- the newer TLDs not at all- so typing "yeah, you can run it too- just go to emby.media (or saying 'embee-dot-media')" will confuse them. However will we survive this tragedy!
I wish I could draw; I keep thinking of a variation on the SF Chronicle's classic "Little Man" for movie reviews. The happy one, of course!