Some clarification. You are considered to be a Jew by birth if your mother is ethnically Jewish. You may be religious or secular, it does not matter. Technically, a person can convert to Judaism (Ivanka Tramp) and become a Jew, but the conversion procedure is very complicated, may take a year or two and is very rare. After conversion, however, you are considered by the religious law (but not always in practice by the community) to be a Jew without any distinction from ethically Jewish.
While this is true in theory, it does not match my experience growing up as son of a Jewish mother and Catholic father.
I was essentially raised as a Catholic, albeit in a community that had many Jews and many Catholics. I was never made to feel "Jewish" by my Jewish friends (of which I had many)- they saw me as a fake Jew and Catholic interloper. They considered me a Goy or a Gentile like my father, and I looked like him, so that was my defined identity. My Catholic friends considered me to be a Jew and not really Catholic, even though I celebrated Easter and Christmas and was even dispatched by family elders to have the food "blessed" at the local Church prior to Christmas Eve and especially Easter feasts. My entire childhood, I never felt like I belonged to any ethnicity nor religion, nor was I particularly made to feel like I belonged by one or the other of these groups. I was not Jewish enough for the Jews, nor was I Catholic enough for the Catholics, nor was I a true Eastern European to the Eastern Europeans, although all of my ancestors on both sides came from Eastern Europe, and I am as Eastern European as anyone is. Simply put, I was made to feel, and actually did feel like I was a half breed, and did not belong to any defined racial or religious group.
Later in life when I dated some biracial women, I heard similar stories of confused racial identities. One very attractive woman I dated, the daughter of a Jamaican father and an Italian mother, told me she never felt accepted by either whites or blacks. She grew up in a largely white community where white people told her things like "you don't talk or act like your black" (an enormously insulting comment she received over and over), while blacks told her she was not really black or was a "cracker" who was an inseparable part of the white community which, considering all her classmates, friends and fellow cheerleaders were all white, she was perceived to be, regardless of her own racial identification.
Indeed, my own Jewish mother wittingly or unwittingly reaffirmed these feelings by encouraging me to view successful half Jews as my role models. Just by way of example my mother, when I was a young child, told me that the actor Paul Newman was half Jewish (his father, not his mother),. Newman then became an early hero in my life to identify with, and I saw all his movies like Butch Cassidy and The Sundance Kid, and Cool Hand Luke:
en.wikipedia.org
Newman was also a revered and highly respected resident in the area where I grew up, that being Fairfield County. I regularly heard stories about him, from various people who had met him, although I never did meet him myself.
So my own mother reinforced that I was a half breed. Even Cloudsurf would probably tell you I am a "Half Breed", just like the actor Zahn McClarnon, who has made a career out of playing Native Americans, but is in fact only half Native American, and gets no credit for his white half:
en.wikipedia.org
But as I grow older I realize that racial and religious purity is being completely decimated by intermarriage between various groups. I think these classifications no longer really matter. As an adult, I would respond to questions about my ethnicity by saying, I am a "mutt." And I said this because of not only the Catholic/Jewish dichotomy, but because I count as ethnically Polish, Slovak, German, Russian, Hungarian, and Israeli, and would arguably qualify (based on heritage) for any of those 6 national soccer teams, as well as for the country of my birth, the U.S.A. I would most want to play for the USA, and then probably Poland or Israel.