Sounds cool, though that's probably the most generic name I've ever seen before 'Game Recording' (though it is pretty clear what it is). Call it something short and hip like, Steam Jetz or some shit.
Whatever their reasoning, I honestly think I'd rather have a simple, intuitive name over a Gamer™ one. Preferences apply, of course, but I often find the latter cringe-worthy.
Without being told what it does "Shadow Play" doesn't sound like a feature that does what it actually does. "Steam Jetz" would have the exact same problem.
Game Recording, on the other hand... It also fits neatly with all the other generically named features like Steam Friends, Steam Community, Steam Achievements, Steam Input, etc.