Apple Music (especially when coming from iTunes) has a very tight concept of a "Library". If you add an album to your library, all the songs are also in your library. If you add a playlist, you have the option to add all the songs to your library.
In my (arguably limited) experience with Spotify Premium, I can't browse all my songs in my "Library". If I'm missing something in the UI, please let me know; I would love to hear otherwise.
I kind of miss the feature that I can upload local files to Spotify (I can, but it is not as seamlessly integrated as I want), but when I "like" an album on Spotify, it is added to my library. I think that the individual songs are not added in the Songs tab. This is a bit different maybe, but in a way, it also keeps that list uncluttered. That might be the problem? So you have the song in your library, you just have to navigate to it via the album.
What I really, really miss is hierarchical filtering/navigating. Genre - Artist - Album. Maybe Release Year. Whatever criteria floats my boat. But so far I have not seen one of the big streaming apps do that, and at the moment I am using Spotify, because it most closely aligns to my UX preferences... but of course YMMV.
In my (arguably limited) experience with Spotify Premium, I can't browse all my songs in my "Library". If I'm missing something in the UI, please let me know; I would love to hear otherwise.