How are economic principles applied to non-financial models, such as a democratic election? Economic agents make decisions in particular models. These models do not need to be related to financial contexts, and economics can be used to describe how agents will act in any other field in which humans act.

As an example, how would economic principles be applied to the process of a democratic election?
Long story short: Preferences orderings are over more abstract outcomes instead of money. It's still a story of maximizing expected utility.

Agents are usually assumed to have preferences over outcomes. This is assuming something like consequentialism. In financial models, the outcomes are dollar amounts and so there is a natural preference ordering. In something like a political model, outcomes are policies. So, people have indirect preferences over political candidates based on beliefs and seek to influence the outcome by voting. This is probably a bad model of voting, as most people don't honestly believe their vote matters I'm guessing.

Beyond voting to influence current decision-making, here is a good paper about voting as communicating: Piketty (2000)