I suspect that outside the US, most Jews rarely if ever think about whether they are "white", much less about whether they feel "white"

US culture is very focused on racial categories – who is and isn't "white". Most other Western cultures are not – European cultures, Australia, etc

I think in Israel, people think about who is Ashkenazi, Sephardi, Mizrahi, Beta Israel, Arab/Palestinian, Druze, Secular vs Religious Zionist vs Haredi, etc, not about whether or not they or others are "white". The question of "who is white" is very American, not one that has a lot of relevance for Israeli society

They probably use some word that differentiates them from everyone else - be it White, Christians, Dutch, Goyim or whatever. In Israel Ashkenazi heavily correlated with appearing "White" btw, and the word White is used when you want to say privileged. So I wouldn't say skin colour isn't a thing there.