It's an interesting issue.
In Wing Commander, you are generally assigned a fighter (and squadron) which will be yours for a fairly lengthy period of time.
This is to say that your average newly minted 2nd Lt. on the TCS Hornet might be assigned to Hornet 608, part of the Fighting Foxes Squadron of FW-188. He will stay there until he's killed, until he's reassigned, killed or his tour ends.
The Tiger's Claw was unusual (and they make a comment about this in WC1) because Halcyon liked to frequently shift pilots around. Also, because it had several 'mixed squadrons' -- containing several of one type of fighters and a pair of Broadswords (for example).
We were able to fly many fighter types off the Concordia because we weren't *really* part of her complement -- Angel simply assigned us to whatever was free. We never got stuck with a single squadron (or fighter). During the addons, of course, we were Special Ops operatives...
(Quick aside: A carrier carries the same number of fighters as pilots -- which is to say that there were 104 pilots on Midway, 400 on Vesuvius, 252 on Midway, etc. The major exception is Cerberus, which had 30 fighters for 6 CIS operatives.)
In Wing Commander III and IV we could fly whatever fighter we wanted because we were the Wing Commander and/or (in some instances) Captain of the ship.
In Prophecy, as a 2nd Lt., we went back to flying whatever was assigned us. Squadrons in Prophecy were organized a bit differently, though...