But it's also synonymous with reverb. The more echo there is, the more likely it is to be "deep". Because it gives the illusion of actual spatial depth in the music. A chord with no reverb will sound flat. Perhaps filtration also adds depth, because again it emphasizes sound as something physical.
Re: emotional sophistication, it can also mean not just emotions complicated, but also emotions that are relatively uncomplicated but are assigned a higher place on the "depth" totem. Melancholy is somewhere near the top. Joyfulness is somewhere near the bottom.