Part 91, assuming this king air does not require an SIC...

Sure, any passenger can do that. The pilot can also allow you to manipulate the flight controls to any degree that he's comfortable with. If the pilot is not a CFI you can't log it as dual but you can still get some hands-on practice.

Also technically...you can log it under "total flight time". Total time is actually not an FAA-defined time-logging category so you can put anything you want there. But if you're career-oriented that type of time logging could create uncomfortable questions with future employers so probably best not to go there.

91K/135/121 it gets more complicated, although small 135 aircraft often have a pax in the co-pilot seat.