There are many possible ways to answer this, and answers are often influenced by people's political point of view.
I would argue that race and ethnicity are so controversial because the US was historically a very racist nation. We, after all, allowed slavery and segregation. We allowed the Chinese Exclusion Act of 1882 and we would not let Japanese immigrants become citizens.
The controversy arises, then, because of competing interpretations of the extent to which that history matters. Some say that it still does, saying that America is still racist and/or that past racism continues to hurt current minorities. But others claim that the racism is all part of the past and that minorities today have equal opportunities. People on both sides get very upset at the other because this can be a very emotional issue.
I would argue that race and ethnicity are controversial in all countries.
Here in the US, there is a long history of racial and ethnic inequality. Many would argue that inequalities still exist today. This is controversial because some groups argue for further work to balance power among the different racial/ethnic groups, and others say no such work is needed. Affirmative action is one example-- many minorities (and women) argue that giving preference is fair to make up for historical nd educational disadvantages, while some majority group members (white and men) argue that that is unfair and claim reverse discrimination.
Other groups call for apologies or financial compensation for historical mistreatment, such as reparation for slavery, lands that were stripped from Native American tribes, or even a formal statement of wrongdoing.
Some see skin color or heritage as a marker of being a "foreigner" and therefore, un-American.
Finally, ethnicity is closely tied to one's religion. We often see conflicts with religious groups as many people are passionate about it.