I think listening is a skill, for many reasons... but there's also different types of listening... Musically? Well, from my years of lessons with my piano professor, one of the biggest things I learned from him was how, exactly, to listen... to pay attention, to listen critically, to 'learn something' from what you're hearing: it's more than just hearing a note played, it's how that note is played, how it plays with another note, all that. The most important part of my warm-ups is to just drop my fingers onto the keys until I achieve 'the sound' - that perfectly balance of striking and bringing out the note, not too loud, not too soft, not too sharp, not too flat.
Many times I would play, especially in my first year or so with him, and it would always sound good to me. But he would comment that it wasn't, and slowly over time, I listened, and he would record me, and as I listened to the music and warm-ups played back, I would learn from them: and eventually I developed the ability to listen to myself playing in real time, adjusting as I play the piece instead of thinking everything sounds great. It's why I have a hard time listening to my old recordings, because I know I wasn't paying attention to the sounds I was making, at least not as much as I should have.
I dunno... I think it is definitely a skill in the context of being a musician, or analyzing a recording. It's also a skill conductors have to have, that is their job, to listen and find the tiniest faults.
With people? Conversation? I'm not a good listener. I don't have the skill to focus, in real time, face-to-face, on someone's problems, whatever they're talking about. I have a very big problem with in person discussion because I am a poor listener. This is why I can't be a sales person, customer support, etc. - they require great social skills, great listening skills, and I can't do that.