I think you overstate the case when you say their background informed their theories and actions. In fact, since they were rebelling against their previous religion, none of the religious ideas or practices informed them and the only thing that remained was the fervor. ‘Fervor’ meaning passionate drive and investment in sweeping systems of thought or belief. Hence the term ‘true believer’ has a strong association with denizens of the left who pursue their political religion with all the fervor of devout theological adherents.
When I speak of background, it is as downtrodden outsiders, not as Jews. You are right, the religion itself does not lead to socialism. But the experience of being a European Jew, and centuries of often unjust and brutal treatment gave them a strong sense of wanting both social justice and maybe a little bit of resentment towards the powers that be. Or am I overgeneralizing?