A lot of the responses here are talking about ethnicity or Jewish identity, but I think when it comes down to it, Jews are considered white mainly when other white people want to consider them as such.
I would bet that your average Ashkenazi Jew (it is worth noting there are multiple different Jewish ethnic groups) has more genetic commonality with an average person of Ukrainian descent than that Ukrainian person has with someone of Irish descent. So why are the Ukrainian and Irish person both white while the Jewish person isn't?
And also let's point out that there is no distinction for any other group of Middle Eastern people. They instead have to choose between being Asian, African, or European.