That's definitely a basic word for a Jewish male, with "Żydówka" will be the feminine version. Note that if it wasn't at the beginning of the sentence, it could be written with a small letter: żyd is a person of the Jewish faith (people of other faiths aren't 'capitalized' as well), while Żyd is a "person with a Jewish cultural or ancestral connection". They obviously overlap very strongly, but it's not 100%.
A citizen of Israel can be called "Izraelczyk/Izraelka".
ok came here to say smth, even tho the comments below show it's been said before, but nothing's been changed. all the Jewish people I know REALLY dislike being called and referred to as "Jew/Jews". and like, even if one Jewish person is okay with it, that doesn't mean they speak for ALL Jewish ppl, lol. it's ok to say "lmao it's all 'phony outrage' if you're the one sitting on top.