Is it possible to train yourself to do so? Or is it something that comes naturally like perfect pitch?

It's a mix. You can train yourself to recognise particular notes. I was always taught to pay attention to the bass in any song/piece because that's your harmonic foundation.

However, getting a sense of the key and being able to follow a bass line and recognise pitches but have a clear sense of "yeah, this is the tonic and it's x note"; this is something that takes time to naturally develop.

I don't know it. When I listen to a song, I usually hear the "name" of the notes in my head. When I play that notes, I know in which key I'm playing. Sometimes I don't need to play the notes to know it. You'll be able to do that with practise. Not only practice with melodies, work also with chords.

Another way to determine the song is to pay attention to the last chord. That chord often determines the key.