In English, it doesn't. The names of those continents are North America and South America. It feels weird to them because they're used to calling the Americas

In English, it doesn't. The names of those continents are North America and South America. It feels weird to them because they're used to calling the Americas "America", the US "Estados Unidos" and Americans "estadounidenses". Well, it seems weird to me as a (United States of) American that they use those names. Our cultures are just different.