If you have access to host the video directly (for example, from your own servers or a Vimeo pro account), the best way to deliver video playback on AEM Mobile Android is to directly link to the .mp4. This can be done either with a video overlay (from InDesign content) or the <video> tag in HTML. Fullscreen video is fully supported from video overlays using the AEM Mobile video player, but not yet from HTML (although it is on our list).
A direct <video> embed tag will use Android's HTML video player, which should recognize that fullscreen is not supported and not include fullscreen player controls. Third party video sites like YouTube use their own players, which may or may not know or respect whether the container supports fullscreen. Some of them are configurable as to whether to include fullscreen controls (I'm not sure about YouTube).