Question

why is america known as us?

Answer 1

Because it is unit of Independent states so it is called United States

Answer 2

This is a topic that leads to huge debates (and often flamewars) online whenever it is brought up.

Logically, it makes perfect sense to use "America" and "Americans" for this country. The name of the country is "United States of America". Why would it be strange to shorten this? It is common to shorten the official name of a country — most people don't even know the official names for the various countries. For example, the official name of Mexico is "los Estados Unidos Mexicanos", which means "the Mexican United States"; nobody is surprised that it is referred to as "Mexico". People would be surprised if you called them the "EUM". (Also, this example shows that even "United States" is not a unique term to one country.)

Australia is officially known as "the Commonwealth of Australia", but we are happy to simply call them "Australians", even though it is also the name of a continent. Depending on how you do your geography, the Australian continent also contains other countries aside from the "COA".

Lastly, I just want to point out that there is no single continent called "America". There is one called "North America" and another called "South America", which are sometimes collectively referred to as "the Americas".

I think the strange thing is not that people from the USA call themselves "Americans", it is actually more strange that the full official name or an acronym is used so often.