The answer is, there was no vassal relationship between these two kings, but a vassal relationship between two kings can happen.

So why wasn’t Henry a vassal of Kings Louis of France? Henry II of England (AKA Henry Plantagenet) was a powerful, sometimes ruthless monarch who was also the Duke of Normandy and Aquitaine, Count of Anjou, Maine, and Nantes, and Lord of Ireland - titles he gained through bloody wars and subjugating entire peoples. The fact that he was the Count of Anjou, simply meant that Anjou fell into the domains of the King of England - and therefore ceased to be French territory. So there was no vassal relationship needed here. Worth noting that Henry happened to be the main rival to King Louis of France, against whom he fought many wars and won, securing the entire western part of France for himself.

However, this isn’t to say vassal relationships can’t work between two kings - they can. But it requires one ruler to be substantially stronger in order to enforce a vassal relationship on the other (that is, one can overrun the country of the other and capture and kill the king there). I’m such a scenario, it is likely that the weaker king is simply a puppet for the stronger one.