Yes, and neither you nor your wife will ever be classified as anything BUT Americans. On the other side of the coin are my brother and sister-in-law, both born here, yet my nieces are classified as Hispanic, WHICH IS A CULTURAL AND LINGUISTIC CLASSIFICATION.
My children and my nieces have retained about as much of a Hispanic cultural identity as you and your wife, yet they are tied to a cultural identity different than yours.
I get your frustration, or even resentment. I've heard related feelings, from others. A cousin by marriage has a Spanish surname. He said it has been his life experience that SOME think he might be an illefal immigrant, because of his name.
He is very conservative, yet expressed disappointment with the nativist branch of so called conservatism (and by extension the GOP), that brands all immigrants, or Hispanic immigrants as probably illegal, and "less than."
His family traces to the early 1800s when California was a colony of Spain, before Mexico.
When people ask me where my surname originates, I like telling the story. I think my Hispanic friends enjoy it too. At least around here, it is okay to be interested in and proud of our roots.
SoCal today is probably among the world's most diverse melting pots. I used to make the mistake of mistaking Asian appearance, with immigration. I know Chinese, Vietnamese, Japanese, Indonesian, etc.
I totally get the thing with first generation immigrants who are strongly proud. Did I say my mother-in-law is one such person? And her brother was, too. BTW he traveled from Italy to visit his sister, who was married via an old world arranged pairing (my mother in law).
He met a young woman while here, fell in love etc. He hung out in Mexico, and came to see her as a short term tourist for awhile. He went to Veneauela because he heard he might get into the states from there. Then he heard it was better to go to Cuba. Apparently he did go to Cuba in the early 50s, and then to the states. Married the woman, lived happily ever after.
They lived very well because she was a doctor, and they went to Italy almost every year to see family and enjoy it.
In his dying days, he stressed how proud he was to be a legal immigrant, and US citizen. Interesting life. Drafted into the Italian Army, taken POW, escaped etc. Some law school before the states.