There are a few parts of this. For one thing there is how Jewish people regard themselves, and then there is how they are regarded by others.

I'm generalizing, but broadly speaking Jews themselves believe that they are a people/nation. One term used among Jews to refer to themselves is "klal Yisrael", meaning "all of Israel." This term long pre-dates the modern creation of the State of Israel in 1948 and is disconnected from any physical location in the country of Israel. E.g. in a letter I read this morning from a New York Hasidic community, the letter used "klal Yisrael" to talk about what was happening to Jews in New York.

To your point about "people who practice their respective religion," notably this concept also applies to secular/atheist Jews as well. Indeed, Zionism, as a project to return the Jewish people to the historic land of Israel and to found a state there, was in large part led by non-religious/atheist Jews, for example David Ben-Gurion. Peoplehood or nationhood is probably closer to the broad Jewish self-conception than "race."

Which brings us to how Jews were perceived by others. The term "racism" is relatively new, and seems to have been coined in French at the end of 19th century by racists to describe their own views. Anti-Semitism, an older term, was the primary animus of this coterie of racist thinkers and was also self-applied. The journalist Édouard Drumont, for example, founded The Antisemitic League of France in 1889. This is hardly a speciality of mine, but while European racists prior to the Holocaust believed all kinds of nonsense about races--(the term racism appears to have been coined by a crank who worked under Drumont in a novel where he expounds on his theory that the Southern French are a different, inferior race than the superior Northern French. He also of course was an anti-Semite.)--Jews were almost inevitably the ur racial "other" in the racist world view. This of course culminated in the Holocaust itself, carried out not only by German Nazis but also abetted by racist anti-Semites across the continent who shared that view. Likewise, to return to the religious question, the Holocaust was carried out without regard to religious belief: A Jew who had converted to Christianity remained a Jew by blood in the eyes of the Nazis.

This is also why many secular/atheist Jews believe that their Jewish identity is identity is important. For many Jews who survived the Holocaust this world view was forcibly thrust upon them. As Stefan Zweig heartbreakingly writes in the opening of The World of Yesterday, the Nazis had stripped him of every other identity that he had:

...I sometimes feel as if I had lived not one but several completely different lives. When I say, without thinking 'my life,' I often find myself instinctively wondering which life. My life before the world wars, before the First or the Second World War, or my life today? Then again I catch myself saying, 'my house', and I am not sure which of my former homes I mean: my house in Bath, my house in Salzburg, my parental home in Vienna? Or I find myself saying that 'at home' we do this or that, by 'we' meaning Austrians, and remember, with a shock, that for some time I have been no more of an Austrian than I am an Englishman or an American; I am no longer organically bound to my native land and I never really fit into any other.

Zweig committed suicide in Brazil in 1942 shortly after he finished the book.

Race is of course, at a basic level a nonsense. There are dark skinned Ethiopian Jews just as there are pale-skinned blonde-haired blue-eyed Jews.

To emphasize the absurdity of this it's perhaps worth noting that under the US census, contrary to the premise of your question, that people of European, Middle Eastern, and North African descent are classified as white:

White – A person having origins in any of the original peoples of Europe, the Middle East, or North Africa.

This question of whether Jews are "white" has also obviously taken on political relevance in the US given the current politics about race. I won't get into that for 20 year purposes, but you can read about that and about some of the history of this question, for example, here: https://www.theatlantic.com/politics/archive/2016/12/are-jews-white/509453/