I've noticed a couple of bugs relating to fleet leaders that may or may not be related to each other. I've been mostly playing as Vasari so I'm not 100% sure if they affect the other two factions as well, but I believe they do.
1) When I select a fleet with multiple capital ships, the ship I've set as the fleet leader is often not the ship that ends up selected (ie, who's abilities I can see on the lower bar in the UI). Instead, it seems that the game sometimes decides to choose a different ship as the "unofficial leader", and that ship is instead the one that ends up selected when I select the fleet.
Eg, for this fleet, the Jarrsul Evacuator is the official leader of the fleet:

However, when I select the fleet, one of the Kortul Devastators (in this case, specifically the one on the left) is the one that ends up selected:

I've noticed this particularly seems to happen when a ship levels up: sometimes, when a ship other than the fleet leader levels up, that ship seems to become the "unofficial leader" of the fleet.
This is annoying because I often set a ship as fleet leader who's abilities I particularly want to micro or control manually, so having a different ship be the one that ends up selected instead defeats this purpose.
2) Sometimes, the game decides that the "unofficial leader" should be the actual leader of the fleet, and switches it to being the fleet leader (ie, the fleet icon starts showing over that ship instead). Not only that, but the game for some reason thinks that the ship must be the fleet leader: when I remove all ships from the fleet, add back the one ship that I want to be the leader, and then add the rest of the ships back, the moment I add the "unofficial leader" ship back to the fleet, that ship takes over as the actual fleet leader again.
In particular, I've noticed this seems to happen most often when I'm adding new capital ships to a fleet, though it doesn't happen every time I add a new ship. And I'm not sure if it's possible for it to happen in other situations as well.
EDIT: Made images bigger since they were kind of hard to see.