Lumia Play to only supports streaming to DLNA renderers (DMR) as they can receive pushed content. It does not detect XBox or PlayStation on your network because they are not DLNA renderers but DLNA players (DMP) that pull content from other DLNA capable devices.
However, there are a few steps you can take to share content with your Xbox, but they do involve a Windows 7 or Media Center capable PC.
You first need to connect your Xbox 360 and your PC via Windows Media Center. The instructions for that are found here: http://support.xbox.com/en-US/xbox-360/audio-video-setup-and-use/windows-media-center. After the setup is complete, start the Media Center Extender functionality from your Xbox via the Xbox Guide button or from any of the Media tabs. Your Xbox will then show up as an available renderer in the Lumia Play to app.