I guess here that in order to build a dark matter detector or to examine ancient minerals to find clues about dark matter, scientists must make some educated guesses with regards to the possible nature of dark matter?

This is correct. Here we assume that dark matter is a weakly-interacting massive particle (aka a WIMP). This is one of the most popular choice for a dark matter candidate, although there exists hundreds of them :)

Are there different schools of thought and political trends with regards to what dark matter might be and, as a result, unlock budgets to conduct investigations in one direction or another?

There are a couple of main paradigms plus really hundreds of different models. The nice point is that the amount of observable signatures is not that large at the end of the day. What is important is that all options are covered. And we hope they are.

Politics does not really matter here. Moreover, with the absence of results, physicists are thinking hard about proposing new ideas.

On the other hand. experiments targeting new options are proposed and some of them are built.