For me personally, I love when albums have a theme to it, like a story or a movie. I like when the tracks on an album sound nostalgic to make me feel something similar that I've felt before, whether that be sound or rhythm. I also love when the lyrics are detailed and talk about who, what, where, when and why cause I can also imagine it in my head and that creates a picture which makes it feel cinematic to me. I also love when albums have perfect instrumental choices for the theme they're going for, like the right sounds and the right tempo and tone of voice for the mood the artist was going for too.
Another thing that has replay value for me is whenever songs have moments, like "a synth lead only comes on during this part" or "this pause only happens here right before the verse" or "the 808 pattern changes here" I feel like those things matter because it will most likely make the listener want to go back and relisten to the song because of that. I think albums should be treated like a movie cause it's also very similar to exposition, rising action, climax, falling action, and conclusion. A few albums I can think of that have done AOTA pretty well is Rodeo, Blonde, CMIYGL, DAMN., I Decided, college dropout, CTRL, and testing just to name a few. I could play those albums front to back with no skips because it has special moments and doesn't feel to repetitive for me personally.
Another thing I love about songs is when it has references to something I know of or when they sample something I've heard before, because then it makes the song more special because I know what the artist was talking about or trying to recreate, same thing with topics as well.
I could go on forever about this but those are a few things that makes songs special to me, that was a good question and topic.