First, I would like to say, that from my perspective (American), I don’t really agree with the statement “In the Usa you are identified first by your ethnic background”, simply because my personal experience doesn’t support that statement. Rarely, do I encounter someone who points out there ethnic/cultural background, unless it is appropriate to the conversation or situation. Also, I know when I have traveled to other countries, I have never clarified myself as being (..........)-American, just American (however, I don’t know if I’ve ever had to state that I was American, most people could tell from my language and such)
However, since I am American, I don’t know how it is perceived by others from other countries (ie you in Britain). So I will try to answer your original question and explain what context I see (........)-American being used and why.
When I or others around me have used their country of ancestry along with American, it has been in the context of identifying their cultural or historical family background. For example, if I tell someone I’m Irish-American, it is in the context of explaining my heritage or why my family has certain traditions that originate from Irish traditions. I find it is the same with Italian-Americans. As an example, it is an Italian tradition (originating from Italy) that cookies are served at weddings. Why do I know this? Because a friend of mine married an Italian-American and when I went to the wedding, there was a discussion about the wedding traditions that his family observed (Italian) and some of the wedding traditions her family observed (German). Never was the references to their family heritages (Italian and German) meant or used as a way of identifying their loyality to a given country or to their race, it was simply a way of honoring their family heritage.
The United States is unique in that many of the citizens living here today have a family heritage that originates from another country and usually more than one other country. I know that I could use more than just “Irish-American” for myself, but once again, I rarely have occation where this type of identification would be appropriate or necessary.