I'm Irish not American or British but i'd guess it's because a huge part of America's identity is that 'we all came from all over the world to unite', so there's no one identity that is accepted as the only default. In Ireland, there are two normals I'd say. Irish names are normal and English names are normal. They're both normal. So I'd imagine in America it's like that but on a much wider scale.
But also, I think it's to do with the aristocracy in the UK, that's still seen as what British people must all be aspiring to, whether they are or they aren't! I think that keeps people in the UK more conventional in their choices.
It's also aspirational in the US.
The names that are identified as 'American' in the UK all tend to have Anglo (or Irish) roots. Blending in in the US tends to be a matter of ditching the Polish, Russian, Greek, Italian, Swedish, German, Irish (as Gaeilge), Norwegian, French, Romanian, Bulgarian, Nigerian, Ghanaian, Chinese, Korean, Hebrew/Yiddish names your family had from time immemorial and striking out into the New World with names that say nothing about your non-WASP origins. The first generation gets called an 'ethnic' name, which gets shortened (Al Capone, for example), and subsequent generations are named according to the local fashion of the day. The Anglo culture is an aspiration and the acceptable default. African American names tend to be mocked because they are defiantly not the Anglo Saxon names others prefer to adopt.
You don't see Polish or Italian surnamey names. It's generally only English, Welsh, Scottish, or anglicised Irish surnames which are adopted. This may be because English is the language of everyday speech and literature (Heathcliff, etc) but the practice itself of using surnames as names is definitely WASPish with a trace of Southern gentleman, and therefore attractive.
The use of Irish surnames is an interesting case though - clearly the surname as name trend is an imitation of WASP practice, but the Irish were shit on the shoes of the WASPs when they first arrived, and using 'Christian' forenames was very much an established Irish tradition when the majority of the ancestors of present day Irish Americans arrived in the US. Irish surnames as first names signifies that the Irish have really arrived in the US.
It's definitely a case of aspiration in the UK. Anything not English/classic is frowned upon because people using names from elsewhere, or those not considered classy names, are clearly thumbing their noses at the British class system, which so many are so heavily invested in that they take the adoption of 'American' naming practices as a personal insult. It's mainly the middle classes who are choosing names like James, William, Charlotte, Ella, etc. These are the people with the most to lose if the class system is abandoned.