The moment you select a solar system you will see its procedural name. In the same way, the moment you arrive at a planet for the first time, you will see its procedural name. Think of it more like a latin name of a plant (sunflowers belong the the genus Helianthus etc.). As you scan a planet, you see those points of interest. Once you discover a required number/percentage of the planet and upload said data to one of the beacons, you'll be given the option of giving the planet your own desired custom name. I'm guessing something similar will be implemented for animal and plant life.
And yes, all of the discoveries and names can be uploaded to the Atlas (a central database, or as Sean describes it, like a collective "pokedex") for everybody to access.
Yeah, that part is interesting and very obscure for now. It's pretty weird to build a server and multiplayer architecture within the game for a feature that will be scarcely or even never used. They are obviously hiding something that might be very important to the lore or crucial gameplay element that we'll need to discover for ourselves, but at the same time they don't want to straight out say that there will not be any chance at all of encountering another player. With that said, even if we do have some sort of Journey-esque multiplayer system, people shouldn't really expect to team up with their friends.
What I mean is, even if you manage to fly to the same planet, the matchmaking system will probably work in a way that might not pair the two of you together in the same instance. Then again, if it's just the two of you (which is a very likely scenario if you manage to pull off the meet), then there won't really be anyone else to pair you up with, so you'll meet each other. The circumstances may change dramatically if there are in fact a lot of people (say 50) in the same solar system or the same planet, and, say, the server will always support a maximum of 5-10 people. Then you'll have no control on who you actually run into, at least not unless you have a friend list that influences this choice.
The chance of meeting up also depends a lot on the tools they give to the player for orientation in the galactic map.
As people have mentioned before, things might get more crowded as people start getting to the center. Still, we don't really know how all of this will turn up, but my guess is that they'd like other players to meet eventually, but to have their interaction severely limited.