Relevantly, I have one Jewish friend who'd never identify as anything but White (and would probably take offense if you labeled him as something else) and one who probably wouldn't consider himself White at all. While the latter definitely has a darker complexion, he's certainly not as dark as someone of Indian descent or even Latino descent.
Yes, some Jews identify as white and others do not, which is true for both Ashkenazim and Mizrahim. But it does not make much sense to categorize Jews, whether Ashkenazi or not, as white. The category white, in the US and elsewhere, does not solely refer to the complexion of one's skin; it refers to a whole set of cultural norms. For instance, someone who would be considered white in Brazil could be considered black in the US.
Indeed, the idea that Ashkenazi Jews are white is something extremely recent. It does not fit history: persecution on the basis of explicitly racialized motives took place until, say, 70 years ago. As late as 1987, the US
legally defined Jews as non-white. Even after WWII,
quotas existed in the US to deny Jews rights US whites enjoyed, and similarly, Jews have never been considered white for most of our history in Europe. It also does not fit biology: Jews do often look differently, and this was certainly the case before the post-WWII waves of intermarriage in the Western world: one could generally tell the difference between Poles and Jews in Poland. Some of my Ashkenazi friends still look so differently compared to Dutch people that they encounter "driving while black" issues. Moreover, scientists concluded that Ashkenazi Jews can be considered
"genetic brothers" of Palestinians, Lebanese and Druze. But most importantly, our traditions and our culture are not Western. They are Middle Eastern. Sure, there has been a lot of cultural contact and exchange between European Jews and non-Jews, but that does not change the fact that our traditions are non-Western. Har Sinai, where we were given the Torah, is in the Middle East. Israel is Middle Eastern. Ashkenazi Jews' "culture" might be influenced by European and US culture, but the core of what Jews have in common is Middle Eastern. Now, of course, most US Ashkenazi Jews have totally assimilated, which has rendered them
culturally white and which will make them look white in others' eyes (either by choice or by simple consequence), but that cultural whiteness would soon vanish if these people were to become more traditional, think from a Jewish framework, and stand up for Jewish causes.
So the argument why Jews are not white is twofold. First, and most importantly, we are from the Middle East and the core of the framework of our traditions (influencing halacha, observance, philosophy, and "way of life") has always been non-Western. Second, we have not been perceived as white for most of our history; the idea that Jews are white is something extremely recent.
Look up the racial numbers of heavily Jewish areas on the US Census.
For example here is a heavily Jewish city in Minnesota, St. Louis Park:
I have never denied the fact that many Jews are categorized or categorize themselves as white in the US census. That, however, does not make it true.