Itch does not have a standard for audio or how to program the game. It's just a store.
Every game you download controls the audio differently depending on how it was programmed and itch.io has no control over this.
The problem must be on your side.
If you are using the itch.io app, try downloading the game from the web and running whitout the itch app. Maybe the problem is that discord is capturing the audio from the itch.io app and not from the game being launched from the itch.io app .