What is the official title of the US government?

I'm not sure what you're asking.

The United States of America is what USA stands for.

Then the different branches have different titles: legislative branch [Congress, which is composed of The House of Representative and The Senate], the Executive branch [The president and the militarily], and the Judaical branch [Federal courts and the Supreme court]

The whole government together is usually referred to as the Federal Government of the people, by the people, for the people. But I don't think that's a title.

I would say The Federal Government of the United States of America