The above answer is very accurate.
However, I would like to add that the idea of separation between religion and politics is quite new and really only emerged with the notion of separation of church and the modern nation-state.
During the time of the Prophet (S), especially after the formation of the Muslim community in Medina, the Prophet handled matters that today would be considered both "religious" (like acts of worship) and "political" (like laws and the military).
This continued to be the situation during the early caliphates and early Arab-Muslim empires as well as the times of the Shi'i Imams; for instance, Imam 'Ali (A) being formally appointed as the caliph and the treaty of Imam Hasan (A). That is to say, their role as religious leaders also involved political matters. Conversely, political leaders such as those who took on the caliphate also saw themselves as leaders of Islam.
One can say a similar thing for many other pre-modern empires as well, which were not led by Muslims.
So for that reason it is not really correct to divide the Battle of Karbala into "religious" or "political" since it involved both. There were clearly matters that today would be considered "political" such as succession (that is, it was not a battle over theology) while at the same time, as the previous response emphasized, it was not a ploy for power or this sort of thing. From the accounts of the Battle of Karbala, it is clear that matters both religious and political were discussed between both sides prior to the outbreak of fighting.
Rather one can say it was a religious objection to the use and assignment of political power, and a political response to it (military attack).