He used to fly KLM cityhopper which is KLM's "regional airline" in US terms, going across most of Europe. Then he transitioned to the 737 because the government bought a 737 BBJ as their new private jet. The government jet is operated by KLM, so the king can fly it as first officer together with a KLM captain.
Fun fact: There is an agreement with the pilots union that the first officer that got replaced by the king operating a flight gets paid as if they flew it.
i don't know about now, but certainly in the 90s you could get a flight from Eindhoven to Schiphol.
i remember trying to get from Eindhoven to London on the phone and being told there were no seats - i'd have to fly to Schiphol (deep gloom on my part). Luckily, when i turned up at check-in, the lovely KLM girl on the desk said "No you don't need to do that - we'll fit you in on the Heathrow flight"
I must say that of all of the airlines i flew with, I probably liked KLM the best
I first read about this in an in-flight magazine; the other story was about the PM who arrived on bicycle and left it outside the palace while paying the King a visit after the election.