I think this is something that varies between people. I was very surprised to learn that my sister doesn't even listen to the lyrics of songs, whereas I do and want to learn them so I can sing along (probably very badly, but hey) and get annoyed if I come to a part where I don't know the words. Likewise if I'm fully engaged during a film I can recall almost all of it, even some time later, whereas my sister can't (or perhaps wasn't as engaged in the examples I have in mind).

I'm sure experience helps too though. When I was younger used to listen to songs from anime and memorise the words despite not knowing the language. I probably wouldn't be as good at picking up lyrics if I wasn't as obsessive about knowing them and didn't listen to the same songs a lot.