Maybe a DLNA server like miniDLNA might be what you are looking for. It does not have a website as an interface, but you are able to stream from it using VLC.
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This one I am not sure about, but maybe SubSonic Advanced might be able to do it. I know video support is mentioned but I don't know if that is video and audio or just the audio of the video.
https://github.com/airsonic-advanced/airsonic-advanced.
Note, it has been almost 3 years since the last update.
I've not used it personally (I use Plex), but I have heard great things about Jellyfin. Does it maybe have the feature you need?
Jellyfin.org
I prefer Jellyfin over Plex, but it doesn't look like it has this feature, yet (https://features.jellyfin.org/posts/533/audio-only-option-on-video-playback)
Are they accepting pull requests?