Was Jesus crucified for his religious claims or political opposition to Rome?
Crucifixion seems to imply that he was executed for stirring rebellion against the Empire, but I also hear it was a Jewish internal affair in which Jesus was punished for his heretical claim to divinity despite Pontius Pilate's counter efforts to spare him, and the Bible verses commanding 'giving to Caesar what is due to Caesar' seems to demonstrate the apolitical/passifistic tone of his teaching. Although it is also said that this is a later historical revision by the Church to condemn the Jews and appeal to its Roman audience, alongside the need of creating a politically correct Christianity in the face of subsequent crackdowns.
Do we have reliable information on the political nature of Jesus' theology and his death in the first century context? Was he a passifistic preacher-shepherd, a Jewish-nationalist messianic revolutionary, or something else?
This might be a short answer, but essentially, there's no reliable information about Jesus specifically that would allow us to make a definitive judgment. The historical context makes it very likely.
Within New Testament studies there is a subfield called "empire studies" or "empire criticism," which branches out from postcolonial methodologies to look specifically at the Roman imperial context of early Christianity, and from that perspective it's really difficult to argue that Jesus wasn't executed for political reasons. All the other known messianic claimants in that period were, and crucifixion was often associated with rebellion. But the best we can do to talk about the historical Jesus is to combine that historical context with close readings of the Gospel narratives, all of which are decades later and obviously adapted toward their authors' agendas. So we can't necessarily take them at face value. But there's not really any information about the historical Jesus that isn't derived from the Gospels, besides one or two things in Paul that aren't relevant to this question.
If you want to look into it more, the big names in empire criticism are Richard Horsley and Warren Carter. I prefer Carter's scholarship, but Horsely has good information, too, and is quite prolific. If you read German, Gabriela Gelardini's massive book Christus Militans has some good discussion.