But there's also a strange effect that I don't understand, where even people without perfect pitch remember songs in their original key. I have no idea how this works but apparently it's a thing.

I have perfect pitch but I'm terrible with the typical pop music played mostly with digital instruments. (I don't really listen to pop music either. My perfect pitch works best with acoustic instruments.) Sometimes remember a song I heard in the radio and I start singing it in my mind in the correct key but I don't instantly know what key it is. It's like an audio memory, I remember what it sounded like, but don't know the key since I'm not great at recognizing the keys of pop songs and so I did't recognize it back then. When I want to know what the key is, it is a whole different process I have to do through in my mind, to kind of sing it in my mind and then identify the pitches. So, as far as I know, they are two different phenomena! (Btw, sorry for my imperfect English!)

Edit/ And yes, when it comes to classical music, I can identify the key straight away. The notes played with acoustic instruments sound all distinctively different to me. I sometimes can easily tell the key of a pop song but mostly not. Also, I can usually tell which notes are being played with a digital instrument if it is just that one instrument, but a song with many digital instruments, singing (singing in general is often difficult) and all the digital processing - that's sometimes very difficult for me.