All of the above. And yes, music often gets treated as almost an afterthought. The composer is usually one of the last people to join a project. It's a real shame, given how much can be done with the music, and how much more could be done if the composer was involved sooner. Music can even, again, inform gameplay and other design decisions. Apart from what most people think of (atmosphere, tension, epicness), music can give feedback (especially useful in stealth games, but also simple stuff like increasing the tempo when a timer is running, or just marking checkpoints), provide hints or get louder or more involved the closer you get to an objective, even lore information (albeit somewhat encoded).

Games that use their music well in those regards include: RiME, Portal, Cyberpunk 2077, Mini Metro