Why do some people get so offended when you identify your ethnicity as "American"?

Four proofs/evidences there is an American ethnicity.....

1. It is an option on the US Census

2. An American would be easily recognized as such when in another country

3. A DNA test can easily match a person to a region in the US due to their mixture

4. The dictionary defintion "the fact or state of belonging to a social group that has a common national or cultural tradition."

There are several reasons people have such strong opinions about whether being an American is an ethnicity or not and they're almost always political or based on jealousy.  There is no specific or official definition of ethnic group.  It's an opinion.  Why are the Irish an ethnic group if they speak English?  If someone says that being an American is not an ethnicity then ask them their justification. 

Ask them if being Mexican is an ethnicity.  If they say 'yes' then remind them that Mexico was colonized by Spain just like the US was colonized by England.  And just like American ancestors came from somewhere else, Mexican's ancestors did too.  Selma Hayak is part Lebanese but she's Mexican. Of course, Mexicans have Aztec ancestry but most Americans who have been here for generations, have Native American ancestry too.

The definition of "ethnicity" is "the fact or state of belonging to a social group that has a common national or cultural tradition."  Being "American" fits that definition.  If some people don't like that, it's too bad.  As you stated, it is a choice on the Census so it must be an ethnicity.  If a white American with no other ethnic connection who speaks English travels to France or China he will immediately be identified as an American even without showing his/her passport.  

The reason it's political and sort of based on jealousy is that the people who have a ethnic identification such as Italian, German or Irish will feel less American if you are telling them that you are American and nothing else.  It also makes recent immigrants such as Asians and Muslims feel left out, although they will be the first to acknowledge the existence of an American ethnic background.

I remember reading about a political activist who said she was the Puerto Rican Irish activist.  Yet she was born in NYC.  Her parents were Puerto Rican and one of her grandparents was from Ireland.  Obviously, she was very proud of all of those connections.  But yet, she did not say she was American even though she was born here.  What does that tell you?  Her connection to those countries overshadows her US affiliation.

Go have sex with your father 

It's you,

Your assigning things incorrectly and it seems like you're trying to trick people. But even you're just dumb, it very much qualifies you as a bigot.

The continent group called America consists of many people who speak English, Spanish, French and Portuguese as a first language. Three of the those languages are based on Roman Latin. Both Spanish and French are still spoken in the US and other parts of North America.

The US national motto is written in "Lingua Latina", and that has nothing to do with Mesoamericans (Mexicans) or Spaniards ... and I'm going to repeat, french is not a German language either.

There is *no* hard historic line between the Latins (Roman Catholic) and Anglos (Anglicans). Yet I'm constantly running into people who say American means Anglo and Latin means South American. But are you really telling me there's no split between Anglos and Latins in the US and they're the same people?

THAT MAKES NO SENSE...EVEN AMERICANS HAVE COME FROM OTHER NATIONS IN THE BEGINNING..MY ancestors came on the Mayflower and I have German in me also..

Because people get offended about stupid stuff all the time.

That being said, "American" is not an ethnicity.   It's a nationality.

It's like asking someone what their favorite vegetable is and they answer bicycle.  It's worthy of minor annoyance and is evidence that they do not understand basic definitions or concepts, but it's certainly not a reason to be offended IMO.

ETA:   If you want to know why you've offended a particular individual, just ask them.   I'm guessing it probably has more to do with your pig-headed attitude combined with your ignorance as opposed to you "identifying as" American.