As for the Jewish side of the matter:
What is going on when a…Jew has faith in their respective God? Is there no way to distinguish, from the inside, whether or not a feeling of faith is valid?
All branches of Judaism have one thing in common: Judaism is not a religion of faith, it is a religion of practice.
While this doesn’t mean that Jews don’t have faith or belief, unlike Christianity which membership and ability to perform is based on faith or belief, Judaism is something someone is generally born into and a religion that is performed (carried out through
mitzvot or the observance of rituals, customs, and practices based on Jewish law), not a religion of creeds or articles of faith.
Many non-theist Jews can be found practicing Judaism, observing kashrut (keeping kosher), keeping Sabbath, observing holy days, even praying. Some Jews who accept God as real will reply “no” to the question: “Do you believe in God?” Why? Because some Jews often reject the Christian view of a deity on a throne that controls everyone’s life and answers all prayers. The Western definition of “God” is often rejected by more than a handful of Jews. To illustrate one reason why: Christianity often has definitive theological answers to tell you what God is whereas you find the opposite in Judaism which will only tell you what God is not.
Since faith is not what drives a Jew or is at the center of a Jew’s religious life as it is in the life of a Christian, your questions about “this feeling” don’t apply. Jews don’t practice their religion on the same basis or for the same reason that Christians take up their religion of Christianity. So you can’t approach it asking the type of questions that only apply to the Christian paradigm.