You can stream music from any provider using your phone, but Apple mostly does not allow third party integration with Siri aside from a few preset commands that apps can register to implement. So using the Siri voice commands you can only control Apple Music, because that's all Apple chose to implement. I think they probably will add support for third party music players through Siri in the future, but Apple is probably not unhappy to give users another reason to choose Apple Music.
Presumably there would have to be some kind of Spotify "app" running on the HomePod, to access Spotify without a phone, but there is no SDK for that yet
It's amazing how so many people have been misled to believe that HomePod is limited to Apple Music, which is not true. I think it's both a failure on Apple's marketing part, and our own fears that Apple can deliberately lock us in to their ecosystem. Not entirely unbelievable unfortunately, but in case of HomePod it's just not true.
Using AirPlay. But as far as I understand from the media coverage, it does not support Home Sharing. So, you cannot say "Hey Siri, play me some King Gizzard and the Wizard Lizard" and expect it to be played from your computer.
Not having to go to your computer and enabling AirPlay and selecting an album is kind of the point of a home assistant ;).
Presumably there would have to be some kind of Spotify "app" running on the HomePod, to access Spotify without a phone, but there is no SDK for that yet