Why does "American" almost always ignore all the non-USA countries on the two 'America' continents?
It doesn't. Russians are certainly counted as European.
People from the "United States of America" are described as "Americans". Similarly, people from the Federated Republic of Brazil are described as "Brazilian". Or if you want to talk about continents, people from South America are often described as South Americans.
So instead of sitting through millions of awkward moments, whomever makes up these things came up with "Asian", which basically can apply to anyone.
I've noticed those from the UK call people from the Indian Subcontinent "asians". I never noticed how they reference people from East Asia.