Strictly speaking, a multi-role pilot woulnd't always be assigned to Alpha wing, bombers are traditionally put into Gamma Wing, for example, and I suspect every destroyer out there has an Alpha Wing, it's just the name of the first squadron designated in combat, so Alpha wing are the first ships in the hangar, Beta are second etc. At least, that's the impression I got.
After all, most Destroyers carry more wings of fighters than there are letters in the Greek Alphabet 
This is something I've actually thought a lot about since I'm trying to make the GTVA seem a bit more military.
I've basically concluded that wing names are assigned in the operations orders for the day. Unless engaged in surge-tempo operations, meaning that either a major offensive is being undertaken on limited resources or the **** has well and truly hit the fan, at any given time a destroyer probably only has a third of its fighter complement deployed. It wouldn't really need to send out more than half under anything like normal circumstances. The remainder are in for maintainance, crew rest, or they're simply not needed at the moment.
Those wings that are designated as "on call" for the day, or whatever period of time, are assigned wing names Alpha through whatever.
In the event of an ad hoc situation developing where wings from multiple destroyers or installations are engaged in a single combat, or wings not designated in the op orders must be deployed, they revert to their internal squadron designations of "Skull Alpha" etc. like we saw in "...But Hate the Traitor".
