It's not essential, but it can greatly enhance the experience. Persona 3's music is the only thing keeping me going so far (I've played like... 10 hours? I dunno. I've passed the first impasse in Tarturus. Also, Tarturus can go jump off a bridge). Yeah, P3's battle music is like in the Top 10 most fitting ever--once you get over the shock of, "this is the battle music?" Star Ocean: The Second Story's music is so iconic for me that I get flashbacks to middle school whenever I hear it, the music made that RPG even better. I love Super Mario RPG's music, especially the boss music, it really gets you in the groove for a boss fight.
Most game music I never actually consider, or realize I like until I hear it much later and remember, "oh yeah, that game--that had good music I guess." Game music hasn't been something I've actively thought about until very recently. I've finally read enough of these threads to actually start paying more attention to game music and finding stuff that I like. Fusion/big band/ska/someothergenresmaybe remixes of the chemical plant theme are amazing, I have found.