But I heard that in some procedures in America you have to say what your ethnicity is?

Not sure what you mean by “procedures,” but no. It’s explicitly forbidden for anyone to discriminate against you based on your race or ethnicity. You won’t be asked if you are black or white or anything else. People might be curious where you are from, especially if you have a strong accent, but you’re free to tell them to fuck off (especially if they are being racist.)

We even heard things like that in some places they divide people by “skin color” and base different cultures on that (we noticed that people from America even speak and behave differently based on whether they identify as “black” or “white”) and even have schools for primarily people with dark skin or so.

We have laws preventing creating neighborhoods or schools that exclude specific races, but there are regional schools in communities that are primarily one race or another. Generations of racial segregation, redlining, blockbusting, and gerrymandering have created persisitent segregation that survives decades after those practices were outlawed.

So no, you’re not misinformed. Historical segregation was real and still affects communities today, but there are no laws enforcing racial segregation. We do have a lot of bigots, as evidenced by the literal Nazi salute at the Trump inauguration today, so you should be careful in places like rural Alabama, Mississippi, Florida, Texas, Utah, and South Carolina. But if you’re visiting a major city, or any famous tourist attractions, you’ll probably not experience any overt racism beyond a Trump bumper sticker.