/voice starts recording on desktop but not on my Android phone (nothing happens). I am the admin of the Mattermost server.
Welcome to the community, Siak! 
The behavior you’re seeing is expected. The Voice Messaging plugin’s /voice works only on web/desktop (mobile isn’t supported per the plugin docs: Voice Messaging Plugin); for voice on Android, use Mattermost Calls instead: Make calls – mobile supported
Yeah, this one confused me too at first but it’s actually expected - the voice plugin (the /voice command) doesn’t work on mobile apps like Android, only on web/desktop. For Android you’ll need to use Calls instead. I’d suggest switching to that or using another workaround for voice messages.