Well we don't run an IQ test based on Qur'an and Hadith even on a Muslim to determine if he/she is a Muslim or not. Just kidding, I mean to say, a community doesn't constantly keep track of ones (a so called Muulim's) activities whether he/she is Praying 5 times or not, fasting or not etc. But all we do as a community to accept one in the Muslim community is see if he/she testifies the Shahada just orally. We don't even know if he/she believes what he/she said or not. But just considering that, we accept one into a Muslim community.
Pretty much of the process remains the same even to accept one as a 'Jew' or not. Again, as I've said, we don't conduct an IQ test based on 'Torah', if he/she scores well enough in that, we'll accept him/her as a 'Jew', or if her fundamental practices are according to 'Torah' (whether it be today's Torah with minor manipulations or the original 'Torah') or not. All we just look into is, if he/she testifies that whether he/she believes in One God or not, whatever name you address God as. All we look is if he/she believes or not. We don't look if he/she is a practicing 'Jew' or not.
Now belief, as we know its a common sense, the 'Jew' needs to testify that "he/she believes there is no God but One God and Prophet
Muhammad Moses is God's messenger"
One's saying of above testimony would suffice to be called as an Islamic 'Jew'. (If he/she believes Muhammad (pbuh) is a messenger of God then he/she is accepted to Islam itself directly)
From here on, I present the support of my above discussion, and if you wanna skip it, you can.
"I had a slave-girl who used to herd sheep for me. One day I discovered that a wolf had killed one of her sheep, and I'm a man from the children of Adam, I get upset like they get upset, and I slapped her in the face. Then I went to the Prophet who impressed upon me the seriousness of my act. I said, 'O Messenger of Allah, should I not set her free?' He said, 'Bring her to me.' He asked her, 'Where is Allah', She said, 'He is above the heavens.' He said, 'Who am I?' She said, 'You are the Messenger of Allah.' He said, 'Free her, for she is a believer.' (Muslim and Abu Dawud).
This is how Prophet Muhammad (pbuh) tested the woman if she was a believer or not. That is what we're supposed to do as Qur'an 60:10 demands us to do.
O you who believe! When believing women come to you as emigrants, examine them, Allah knows best as to their Faith, then if you ascertain that they are true believers, send them not back to the disbelievers, they are not lawful (wives) for the disbelievers...... That is the Judgement of Allah. He judges between you. And Allah is All-Knowing, All-Wise. (Qur'an 60:10)
How will you examine them, by conducting an IQ test on Qur'an? Nope, by asking two simple questions as Prophet Muhammad (pbuh) did above in the Hadith. The pretty much remains the same whether to test if a person is Jew or not.
Ask him/her simple questions (not on the entire Torah) that whether his/her definition of God match with the true expected definitions or not. (e.g although if he/she believes there is only God, but what if she believes that if that One God is some Moon God, nope, test failed either reject or help him/her to improvise. Its on you), and some simple questions to test whether if he/she believes that Prophet Moses (pbuh) was a messenger of God or not. If, you're satisfied with his/her explanation, Good, accept him/her as 'Jew', else either reject or help him/her improvise.
Note: This answer attempts to answer "Who is Jew" in general. But this in no way answers whether if a woman can be considered for marriage or not. There goes another discussion for it which would mostly constitute recommendation, although the true technicality remains the same as discussed above. Also this answer presents pretty much the same thing as mentioned by others i.e. Fatemeh, green4rrow in their answers
May the creator guide us all.