muu muu and shift are different styles to me.

A muu muu is a housedress - loose, a yoke with a body gathered off the yoke. A shift is more form fitting - one piece from shoulder to hem.
I've made both... wore shifts in my teens and into twenties. Have a muu muu now, but I don't wear it, too much fabric for me... BUT, it is cool, as related to heat/cold, open to circulation.