A lot of Jews changed their “Jewish sounding” names to not stand out. Many Jews were/are fearful of insults, attacks, and discrimination so it is an attempt to ‘blend in’. Not uncommon at all.
As far as ‘identifying as Jews’, there are a lot of things going on here. First, the intermarriage of the father to a non-Jew, and second the desire to remain connected to one’s Jewish heritage. Also, not uncommon in the diaspora. The reform movement and the conservative movement make their business to teach intermarried couples that the children of non-Jewish women are Jewish, merely because they desire it to be so. If it was an issue, it would invalidate their conversions. Hocus pocus, your kid is a Jew. It is also very common in America.
In essence, Giffords represents many Jews in this country who are very much lost to their Jewish identity and yet still wish to be called a Jew. Demand to be called a Jew. I have been in long and frustrating discussions on FR about this. Non-traditional Jews have much invested in this point of view. It is a no-win argument. Every time.
My Great Grandfather and Grandfather who both had very jewish sounding first names but not last names purchased a business in 1916 that was named after an Anglo family name. That business today still has the name some 95 years later and they didn’t change it in 1916 as they didn’t want it to sound too Jewish. The threat of anti-semitism has always been there so it was never changed. Today that is not the case but the mindset in the 1916 to 1980 time frame was such that it was best not to make any waves. Today we keep the name as its a very established company that is over 102 years old.
If I had a name like Akiba Hornstein and lived in the wild west then I would probably changed my name to fit in and run a business.