Yeah... I'm pretty sure there is no such thing as "South and North America" it's just one big continent called America, and then there's The united states that call their citizens "americans" instead of calling themselves staters or some other shit.

Actually some countries consider North and South America just one country and call it America
ex:one of those countries its mexico

But here in the US we split the continent in two, but we still as a whole call it the Americas.

Not really. It's not like (guessing you're referring to) the UK which is made up of individual countries that are in some kind of alliance.
We're not in an "alliance." That would mean we could break out of that alliance whenever we wanted to. We're individual states that all agreed to be part of the United States. The thing is each of the original colonies were founded for different reasons and only permitted certain people to live there. The United States made them all part of one country with a Senate and a House of Representatives, that allows the states to coincide and agree on a federal level. The States still didn't want to forfeit their right to govern themselves when they entered the Union, but they are still subjected under Federal law, which overrides State law.

It's like the Roman Republic, which is what they tried to base the US Republic after. This is also why the US uses the Eagle as it's symbol and Washington D.C has Greek and Roman like architecture

No, no its not...