You can record game plays with the inbuilt screen recorder. It can be accessed under Settings -> Advanced features -> Game Options. Just turn it on and record the gameplay. Now if you want to make it as easy as it's on iPhone and record any app function, you need to use third-party apps from the play store. Here are some good apps to record your android phone screen. You can use any of them to record your screen. I would use an app which gives a widget for faster access and keeps in on one of the home screens. It's not as fast as accessing control panel but that's the closest we can get.
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