People from those continents call themselves south americans or north americans. Just as you have european and asians, people don't call themselves eurasian.
Furthermore, both people from north and south america understand that american means a person from the USA. It doesn't refer to a person that may be from either one continent or the other. Also people don't call themselves western hemispherian or eastern hemispherian either.
What does African refer to? a person from Nigeria and Zimbabwe or also a person from Egypt or Algeria, or are they north-africans? Are algerian-americans african-americans?
Who is a 'new yorker', someone from manhattan, or new york city, or new york state?
Furthermore, both people from north and south america understand that american means a person from the USA. It doesn't refer to a person that may be from either one continent or the other. Also people don't call themselves western hemispherian or eastern hemispherian either.
What does African refer to? a person from Nigeria and Zimbabwe or also a person from Egypt or Algeria, or are they north-africans? Are algerian-americans african-americans?
Who is a 'new yorker', someone from manhattan, or new york city, or new york state?
Sometimes things are blurry.