The reason Azure Media Player works in your scenario is that it does browser detection to determine video playback capabilities and then use the appropriate adaptive streaming protocol (HLS, MPEG-DASH, or Smooth Streaming) that work with those capabilities. AMP understands the Azure Media Services format tag, as documented on https://learn.microsoft.com/en-us/azure/media-services/previous/media-services-deliver-content-overview#URLs. However, for other players you would need to explicitly set the format tag. If you use a third party player typically you will need to specify a specific protocol since the Azure Media Services format tag is Media Services specific. You may need to do browser detection depending on the player you use to pass it the appropriate URL.
Shaka Player lists both the 'Platform and browser support matrix' as well as the 'DRM support matrix' on https://github.com/google/shaka-player. Incidentally, Azure Media Player has a similar compatibility matrix on https://learn.microsoft.com/en-us/azure/media-services/azure-media-player/azure-media-player-playback-technology#compatibility-matrix. There are also some good docs at https://github.com/google/shaka-player/blob/master/docs/tutorials/drm-config.md on how to do DRM with Shaka on the Shaka side and https://learn.microsoft.com/en-us/azure/media-services/latest/player-shaka-player-how-to on the Azure Media Services side.