Does each separate “organ” here have any relationship to the two separate functions identified in “Thinking, Fast and Slow”?

In short, no. The two types of thinking are both forebrain activity. One comes to a quick guess with pattern matching, the other goes slower with reasoning. It could be that some fast thinking is influenced by the hindbrain in some way, but the thoughts themselves are constructs in the conscious mind that inhabits the frontal lobes.