This ability has been missing for quite some time. The main way to stream music directly from another device to an Echo is to use Bluetooth, and that will still work. Alexa Cast is a welcome new option, though, and the feature is available now on the iOS and Android Amazon Music apps. Just be sure to update the app to the latest version.
Turning on Alexa Cast is not very different from using Google Cast: Simply hit the Alexa Cast icon and choose the Echo device where you want your music to play. The app will stay in sync with the music as it’s playing through the Echo, allowing you to stop playback or control the volume quickly.