Ahh, I see. I was thinking Spotify was saving a reference to a Song, an Album, or an Artist rather than saving a list of songs and showing Albums and Artists from the list of Songs that have been saved. Kind of a bummer really. I think saving an Artist would have been handy. Especially considering the relatively low number of songs that can be saved (10k).
Specifically, if you could save an Artist, it would be nice to have the discography listed dynamically so when a new album is released you don't have to go search for it to find it.
Try this to save an artist: play any song from an album by this artist, then click on the +-icon in the lower left corner where you find the album, artist and cover info (see attached screenshot). The artist is then added to your artists list and the icon changes appropriately.