I know you are stating this is a Ship building game, but what about Space Stations? Could be used to introduce people to building and balancing functions.
The focus of the game is on the crews and the missions through the filter of the ships you build. Each ship and crew have a story to tell, and missions to go on.
In my mind, asking about fleets of ships is the same about asking about neighborhoods of houses in The Sims. Coordinating several ships at once seems like it'd make it hard to feel each ship's story, but there's nothing against having several ships, switching between them, moving crew between them, and so on. Like in The Sims: you don't manage two houses at once, that'd be a mess, but you can throw a house party or go visit.
If you build a hospital ship, you're going to be taking on medical missions - usually several at once. If you have a cargo transport, you may see the opportunity to take on a medical mission, but then why not pass that on to the medical ship? Why not take a vacation on a space station that you also built? Why not run cargo to a planetary base that your friend built? While you're there, why not interact with that crew?
I don't have concrete plans for how to handle multiple ships, but I can't help but turn towards Star Trek again. Most of the time, it was just one or two ships involved in the plot. Any time there were more ships than that, something incredibly huge was going down and the individual crews weren't as important any more.
Space stations are certainly possible, they're really just a cut-back version of starships. However, since they require vast amounts of hab space, there is a question about how to display them. Space stations are probably stacked vertically, and when you want to see who is in which rooms, that can be difficult to display. Hm!