He isn’t part of the royal family. The picture cards are also known in English as the ‘court cards’, so he’s a member of the royal household, yes, but not of the family.

‘Jack’ in the 16th and 17th centuries meant a male servant without a specific role or skill; so not a cook, gardener, coachman etc, just a strapping fellow who can carry stuff, be sent on errands or brought along as a guard, whatever you happen to need doing. The French word for the card is valet, which in French means exactly the same thing.

The other English word for the card is knave, which originally meant ‘boy, young man’, as its German equivalent, Knabe, still does. It evolved to mean ‘young manservant or henchman’. Eventually it became a derogatory word because royal households had so many of these young men, who went swaggering around the streets picking fights, molesting the girls and generally making nuisances of themselves.