One very important reason to stress tha Giffords is not a Jew is that Israel these days is being flooded with Gentiles,who are considered "jewish" by Israeli immigration laws. These immigration laws are emotionally based on the Nazi Nuremberg laws which defined someone as Jewish if he had a Jewish grandfather, without regard to halacha.
The time has come that at least the true Jewish right starts to educate the world that a Jew is a Jew according to the standards of halacha and not according to the standards of anti-semites or not because some Gentile proclaims that he has some ethnic or cultural affiliation with Judaism.