Stream machine is a console, controller and everything. Though yeah in shooters you'll get destroyed by the users using mouse and keyboard.
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It's both really. It's built on PC hardware, like Xbox one and ps4, however instead of running on a unique console OS it runs Linux, but still supports any controller you wish. What this does mean though is that you will be connecting to the same game servers as PC users. So if you were to play COD on steam box you'd be playing vs PC users with mouse and keyboard and would get owned. COD is probably a bad example as I'm pretty sure it doesn't run on Linux...