The truth here is very simple. Our ability to produce constantly expands over time, and the end result is that people today have a lot more stuff than people in the past did. Older homes didn't even need a closets at all. They used wardrobes, chests of drawers, and hooks and begs for their stuff. As the amount of our stuff expanded, so did our places to keep our stuff. Newer homes, have larger closets.
England, and most of Europe, with it's long history, is full of very old cities will small streets, small homes, and small closets. The US, with it's very short history and massive amounts of land, is full of new cities, with wide streets, big homes, and big closets.
In addition, the US has a population density of 83 people per square. The UK has a density of 660 people per sq mile. More land per person, bigger homes, bigger buildings, bigger closets.