For one, white supremacists don't care what Jews look like: No matter what, they don't consider them white. While I don't particularly care for the opinions of such people on this matter, it does reflect that the history of Jews is noticeably non-white, and that Europeans for millennia didn't consider them white or anything like them. While the origin of that bigotry had no scientific basis, there is genetic evidence to support a Middle Eastern origin for Jewish ethnic groups no matter where they are. And yes, it specifically highlights the Levant area, placing their origins in Israel itself. If you consider that area non-white, then you shouldn't consider the Jews white as the influence of European genetics on them is actually found to be minimal. And that's not even considering the huge amount of Jews from families who never left the Middle East, for example Yemenite Jews who were rather isolated from the rest of the world. Of course, some people do consider these Middle Easterners white, and had to put that on official forms until recently, which only highlights how subjective the terminology actually is.
Now, a lot of Jews (including the "European" ones) are actually rather dark. They may still code as white, but the tanner variety akin to Greeks or Southern Italians despite living in Russia or Germany. Jewish ethnic groups actually display a wide variety of skin tones, which is obvious in my family if you compare my mother (a fully Ashkenazic woman who is mistaken as Arab or Turkish often) with her biological niece, who is fair skinned and blonde. For the whiter among us, one does have to assume intermarriage and that's also shown in genetic studies of the Ashkenazi Jews. There is still a stronger Middle Eastern contribution to their gene pool, but we still see a noticeable amount of those genes in the pool. That cousin I mention is an example, as she's half-Ashkenazi and half-Irish biologically, with a convert mother. On a more darker note, there are also those who suspect that a lot of the whiteness also comes from cases of rape, as during the pogroms that fate was often inflicted onto females unfortunate to be there at those times.
This wide spectrum of potential skin tones isn't unique to the "European" Ashkenazi Jews, though. We see something similar among Arab populations, particularly in the Levant around Israel. Despite what is displayed by the news media, you cannot determine the difference between an Israeli Jew and a Palestinian Arab by skin tone alone. The Israelis know better, especially since so many of them would be classified as "Palestinian Arab" by such a schema. Ditto for Palestinian Arabs who look whiter, and can pass as "Jewish" by those who don't know better. Even within other Jewish populations, there can be a lot of variation where some look dark whereas others are lighter in skin tone. I've seen this in Persian Jewish families as well, some of them passing at least as Ashkenazi though others can look rather European as well. These families had 0 contact with Europe until now, which just shows that things are a bit more complicated that many people assume.