On the phone android google playstore app, the battery life of the connected stadia controller, is not visible. Altho, reconnecting the controller icon+y might show the battery life in detected bluetooth signals, during the apps search for nearby controllers, perhaps.
@TwinkleStar Yes, I've noticed this too. It looks like the controller battery level is only displayed in Stadia on Chromecast or Chrome, but not the mobile apps.
Please be sure to use the built-in feedback option to let the developers know and encourage them to address this in the future.