As I said before, I don't know the names of Obama's daughters but I have a vague idea they're out there. Its just not in my choice of interests to know more about them.
You raise an interesting point, and I can see what you mean. I don't know the name of the children of my country's PM or of the GG.
But, Harry and other royals are a bit different from other children/grandchildren of a head of state/head of state type person.
Most elected politicians are in the news for a limited time. Attention towards their families is limited - particularly outside of elections and scandals (when they're more scrutinized). Knowing the names of the children of Obama, George W, Clinton, etc, is less likely to occur because after eight (at the most) years these families typically begin to fade into the background. Their are exceptions of course; political families who are more dynastic in nature and thus whose names are more likely to register. People are likely to be able to list more Kennedys than Obamas.
With royals, however, the relationship and media exposure is different. The press didn't wait outside a hospital to get a first glimpse of Obama's children when they were born. Heck, even now I doubt they would do so if a president's wife gave birth. They did do that when Harry was born. They did take note of his various milestones, and scandals, and will very likely continue to do so for the rest of his life. This is a more normal relationship between the media, general public, and a monarchy.
I would expect that the vast majority of the British are able to identify HM, the DoE, Charles, Camilla, the Cambridges, and Harry. I would think that fewer younger people who aren't overly interested in royals would be able to identify Anne, the Yorks, or Edward (and even fewer would be able to identify Anne's children (or grandchildren), or Edward's family), but older non-royal watchers probably do.
As to the whole "people don't know who Prince Henry is" thing, that's different from people not knowing who Prince Harry is. Not knowing what his actual name is doesn't mean that you don't know who he is, since he's referred to as Harry almost exclusively within the media.
We're not taught the names and members of the royal family at school anymore, they're not of interest to people unless they do something outrageous. When Henry marries he'll get coverage, but people have lives to lead and unless you're a fan of royalty it's the celebrities that get people talking. If I wasn't a royal watcher, I probably wouldn't know who he was.
