Why has the United States' forum for political representation both at federal and state levels never been called 'Parliament' but instead 'Congress'? Does it have to do with anti-British sentiment in the early years of the U.S.?
Interesting question. In eighteenth century British English, a "congress" was a meeting or gathering; in Samuel Johnson's Dictionary the second definition was "An appointed meeting for settlement of affairs between different nations." ("Parliament" was a proper name, not often used as a general noun.) When the first Continental Congress met in Philadelphia in the fall of 1774, it was not a government body, it was just a meeting of representatives from the different colonies, some elected by legislatures, others from revolutionary "committees of correspondence." Those at the meeting did not yet agree on aims, war had not yet broken out. Successive "congresses" carried the name forward; it was in the Declaration of Independence that the name becomes a "style" for an existing group that declares itself to be a government--"The Congress of the United States of America." This language is still deliberately vague, since the delegates don't agree on whether the United States is a single nation or a confederacy of thirteen, so it is still a "congress" in Johnson's sense. After Independence, the new states adopted various terms for their elected bodies--Virginia's House of Burgesses became the House of Delegates, for instance; Massachusetts entities became "courts." Finally, the draftsmen of the federal Constitution carry the same term forward without discussion, partly one supposes because they have been authorized only to improve the Articles of Confederation. Undoubtedly you are right that there was also a sense that a new republican form of government ought to have its own language--other terms like "citizen" and even "constitution" were new Americanisms. (We could use some good secondary materials on this topic; any suggestions out there? I am working on a study of early citizenship, see for instance ssrn.com/abstract=2044805.)
The word "congress" existed long before the US, and it means to meet together. In this case it was a body being put together from several nearly independent states. It expresses the desire for unity without necessarily subordinating the states.
I've never heard it called Congress at the state level. It is usually called the Legislature or Assembly or in some New England states General Court.