That differs from my understanding: I got the impression that once a player explored a world, the parameters for procedural generation on that world were fixed and uploaded (this is an MMO, whether or not the game designers want it to be thought of as such). That's why in the gameplay videos, you see onscreen tags identifying which player discovered a species or world.
I've worked on my own idea of a procedurally-generated universe, and the idea I've come up with is you generate a random list of stars and assign a seed value to each star. When a player visits the star, the details of the stellar system (which are only observable up close) are procedurally generated using that star's seed: planets, asteroid fields, etc. Anything you do to modify that system is saved so that the next time you go back, it's still there. In a multiplayer environment, any modifications or even the results of pseudo-random generation can be uploaded so other players see the same thing you see.
So it's random in the sense of "not determined ahead of time." The way stuff is generated in this game is the developers create prototypes that have traits that can be modified (see dev discussion of character creators with sliders in other games) in many ways, and when a world is first visited by a player, which animal prototypes are present and how they're modified is determined at that time. What the devs are doing by reviewing the results of this generation with bots and gifs is ensuring the range of parameters for pseudo-random generation is acceptable, i.e. you get results that look good. They're not setting the parameters for each planet you'll visit when you get your copy of the game.
The game universe is the same for everyone. The devs are generating it and when they are happy with the way it turns out, that version will be the one they release for everyone to play in.
The universe is the same for everyone because randomly-generated content is preserved online. The devs are not generating the entire universe, they're fine-tuning the parameters of their generators.
As far as I can tell, anyway. This could be really cool. I've also been following the development of I-Novae Studios' game engine, which aims for realism.