In the Star Army of Yamatai, expeditionary fleets are fleets that are made to go on expeditions far outside of Yamataian space. The First Expeditionary Fleet, for example, was created to travel to Ayenee and other universes. The Fifth Expeditionary Fleet was created to go to a long-closed RP site called Galaxy's Edge. The mission of expeditionary fleets is currently to each go to a new sector in the Kagami Galaxy. Because of expeditionary fleets are often at least partially isolated from the YSE, they have different logistics setups than standard fleets (carrying their own farm ships and so forth).
The Star Army's standard fleets are based on a combination of geographic area and function. They used to be standardized (hence the name) but lately the First Fleet has been taking over as the main "everyday" fleet of the Star Army and has grown enormously while the Second Fleet has also grown, but more as the strategic fleet that conducts big operations like fighting off invasions or doing major operations. Other smaller fleets have claimed their own niches to justify their existence so they don't get merged into the first and second.
In the Star Army, the fleets don't really interact with either other more than they have to. Ships have their captain, and their captains report to Shosho and Chujo who report to the Taisho. The Taisho reports to Yui. That's the chain of command and other fleets are not able to interfere with a fleet's chain of command unless invited to. For example if YSS Resurgence wants a Second Fleet Ship to move out of a docking bay, it can either ask that captain very nicely or it can send a request up its chain of command which then goes down that ship's chain of command, hopefully in the form of orders to move. When fleets do want to make a mixed unit with ships from multiple fleets, they typically form a numbered task force for the duration of the mission.