serious question
Is being Jewish something that one inherits or is it a religious belief?
Complicated questions deserve complicated but thorough answers.
The Jewish religion was the societal foundation of the Hebrew people, and as a result, sometimes the word Jew is used to refer to a person of Hebrew ancestry. But this is technically a misnomer and the two should not be interchanged so freely. The process of conversion to Judaism seems to illustrate this point: one need not have Hebrew ancestry to convert, and the word used to describe such a person is ger...but a person who has Hebrew ancestry in their maternal lineage is considered a Jew already so no conversion is required, and this person is not a ger. It was a way to keep the Jewish faith closely attached to the Hebrew people while remaining open to all.
Judaism is a religion, and one observant of it is a Jew, regardless of his DNA. Hebrews (especially ahkenazim) have unique identifying DNA markers, so genetically someone can be Hebrew. Judaism accepts converts, including non-Hebrew individuals. But a Hebrew can be an atheist or a Buddhist.
So...to recap...not all Jews are Hebrew, and not every Hebrew is a Jew.
Jews: adherents of Judaism
Hebrews: those with ancestry that includes the ethnic group historically associated with Judaism.