I currently volunteer at an Animal Shelter during my free time.
Well thank you for doing that.  It's kind of nice how people have changed the name pound, to animal shelter, to change the thought process associated with the word.
Yes it definitely sounds better!
I believe the phrase was coined when animals or stray livestock would have to be penned or "impounded" until claimed by whoever owned them.
That makes sense. When did animal shelters come to exist?
Systems like shelters/pounds have been around a long time, however institutions like the Humane Society have only been around since the fifties when it was founded in Washington D.C.