Interesting question - and in fact many Americans do give their guns names - there is even an article in the American Rifleman from June 14, 2011 titled "Why Do We name Our Guns".
What may be unique is insisting that they all need to be named after a female. While it seems to be traditional to name machinery (almost any machinery will do - from guns to boats to cars to planes) with a female name, the naming of guns may be the most varied - some are named after females, some may be named for beloved pets or hunting dogs, many are named for sentimental purposes, and some may be named for their destructive capabilities (death stroke, etc - though I really have to wonder if we want people who name their guns things like death stroke to actually own guns - seems to me there might be a bit of mental instability amongst those folks - I'd much rather hunt with someone who names their gun Claire or Blue of Old Yeller than I would with someone who names their gun Vengeance).
Is it uniquely American? That might be hard to answer - none of us on this side of the pond have any experience in the British armed forces so for all we know, British Marines name their guns. You also have much stricter gun control laws over on your side of the pond so it's less likely that someone will have a gun.
Or it could just be that we Americans are just as eccentric as many Brits are - just about different things. I have a rifle and a shotgun at home - the rifle is a Remington, the shotgun is a Winchester. I've named my rifle Winchester and have named my shotgun Remington. I'll bet I can find some folks in England that have an encyclopedic knowledge of Dr. Who and who can tell us some really weird things about sonic screwdrivers.