I know that players will be able to share ships and crews, for sure. It should also be relatively easy for players to have mission outcomes mirrored across universes, allowing them to affect the same galaxy... but they wouldn't necessarily be able to see each other.
I don't know how much realtime multiplayer there will be. Realtime multiplayer is a bit tough to do. I'm building the framework to be as "network agnostic" as possible, so I'll take a stab at adding in live multiplayer later in the dev cycle... but in honesty, that sounds like a "Galactic Line 2: Electric Boogaloo" feature.
Regarding uniqueness:
The reason your "slightly faster, slightly poorer welding" ship feels boring and samey is because it's being measured in terms of those flat stats. Rather than make the stats ever more complex, I want to measure your ship's uniqueness in other ways. Specifically, in terms of the missions it takes, the crew it grows, and the history it has.
For example, as missions come along you'll find that a faster ship has opportunities to accomplish missions about racing from place to place, while a ship that has a more effective repair crew will find opportunities to accomplish missions featuring larger or more devastating damage.
The two will slowly diverge as you take missions your ship feels comfortable with. In a few months, the fast ship will be rushing from crisis to crisis, perhaps even serving as a courier or mid-battle repairer. The slow but more effective ship will be waist-deep in the most devastated sectors, helping to haul whole settlements out of a disastrous asteroid rain...
Would those two ships feel distinct from each other?