Personally, I think that it is possible to live one's OWN American Dream. Yes, there are ways that the American Dream is defined as, but that does not necessarily mean everyone's "dream" is the same. For example, an immigrant's dream might be to buy a home and have a steady job in the U.S.... that dream is achievable. Someone else's dream might be a bit more far-fetched and un achievable in their circumstances.
Hope this helped <3