When a piece of music is created (in a fixed medium), it gains a copyright, generally granted to the creator(though this right can and often is traded, sometimes before creation).
Streaming services, such as Spotify, buy licenses for the music that they stream. Their APIs are means to access this streaming service on multiple platforms (e.g. computer, smart phone, etc).
It is almost certain* that the license does not allow for the creation of "derivative works" of the copyrighted work, which a video game would be. Thus, streaming services do not have the right to give you API access for a game (and why would they want to, even if they did have that right)?
By explicitly denying you the ability to so use their service, the streaming services are a) not breaching their license, and b) attempting to indemnify themselves in the case you do somehow use their API to play copyrighted music in your game.
*The license would have to explicitly allow for the creation of derivative works; the copyright holders wouldn't give it away for free, because some game developers, such as the makers of the Guitar Hero and Rock Band series of games, would be willing to pay for them; as far as I know, there is not a streaming service producing a rhythm game, so it would not be in the streaming services' interest to pay for such an extension of their license.