Wanted to give other people a chance to respond before I shared my own stance on this, but I'm actually somewhat in between. ( @Goblin I did try editing the poll after to include a "sometimes" option but don't know how to do that. Anyway, getting more polar responses to a poll is usually more interesting when it comes to light-hearted topics.)
I care about other people's music/playlists and will listen to them only if I actually like their taste in music.
There are genres of music that I just can't relate to and give me a headache. In those cases I tend to find it a bit annoying if they share tracks my way, and it feels like a chore to listen so as to be a good friend. I'd rather just read the lyrics if it's a character I care about.
If we have the same taste in music, though, I'll happily listen to their entire playlist, even if it's 100 songs long and we've only RPed once.
I find it can often be a good bonding experience on an OOC level, too. If someone posts a Gin Wigmore song for their character I get all excited and want to hit them up to fangirl together. I've sometimes specifically sought out characters that I otherwise would not have thought to approach just because their playlist made me think they seem interesting and fun. (A choice played-by also has this effect sometimes; I don't care how "hot" the played-by is, but if it's an obscure played-by from a film/show I really like or something it'll get me excited.)
With this caveat in mind, people sharing a tracklist for their character does enhance my experience/enjoyment/understanding of their RP.
Just don't be that bozo who sets their music to autoplay or something, this isn't the 90s.
For extra bonus points if we have the same taste in music and you link me a song that makes you think of my character and/or our characters' dynamic, you are now my best friend for the day. (I had an IC romance going on with a fellow Hozier fangirl once, it was fucking magical. She sent me Take Me To Church and I cri everytime.)