Honestly, if they wanted to push Linux gaming, this was THE perfect way to do it.
PS4 running on Linux and a SteamOS running on Linux = a giant surge in Linux based gaming.
Nope, not really you see. The systems are very similar but very different at the same time.
Wheras Valve has the Linux kernel, and I assume their own environment to that PS4 has FreeBSD.
Now to compare these to in easy terms, prepare for a bad analogy.
Imagin you have a Car engine and a Car.
With Linux it's like you have the engine only(kernel) and all the other parts, wheels, body, etc etc aren't included wheras with FreeBSD you get the whole car.
And FreeBSD is really slow with their updates, you aren't allowed to submit your own code directly but instead you have to through a "registered" committer who will review your code and then submit it whereas in Linux, go Gungho dude, make and distribute your own Kernel if you want.
Anyone can modify FreeBSD to their pleasing but that is only for internal usage then, if they want those uses to be released they will have to go through that mentioned Committer.
So in other words, yes very much alike.
But FreeBSD is a complete OS,install from source, easy to modify to your own Tastes.
Linux, just the kernel mod as you want and release, rest of the OS, go Gungho and instal/mod and use any libs, install from any repo/source/whatever to try and make a complete OS.
Sorry for bad explanation, you can probably google a better answer, but remember that
it's not the same.