White Star Line may well have been owned by America at the time, but it was still a British ship, with British officers, British insurance and under British Board of Trade regulations.
In this day and age of computers, here in Grim Britain, (I assume most/all others are the same) there is only one port of registry- Cardiff (offices located there for political reasons). Any port name painted on a ship now (UK) such as Dover, London, Birmingham (
) is purely for historic/ image reasons.
In the past, boats and ships were all registered at their local port and carried the port registration and port name for identification. And seems WSL was based in liverpool, it seems a good place to register the ship.
Incidentally, why they used Southampton is a mystery to me. Liverpool is much less distance- was Cherbourg REALLY that important to them?