Is there anything available in 4e that works like Leadership did in 3e, allowing a player to take on an NPC henchman?
No. There is no mechanical choice that allows you to bring along a NPC regardless of circumstances.
However, like in all editions, there is "good role-playing". In other words, if it makes sense for the King to send along a paladin caretaker to protect and guide you, it might just happen.
I have a distinct memory of at least one Dragon adventure featuring an NPC willing to accompany the heroes as a fully-featured party member (including fighting), but I can't remember what it was about. There was no special rules (that I can remember) except the good old "don't let your NPCs overshadow the PCs" DM advice.
In other words, as a mechanical (player-controlled) feature, this is way too over-powered to ever be allowable in 4E.
But as a purely role-playing driven event (read "DM controlled"), it seems WotC handles it fine; just as about every other rpg does.
Think of it like artefacts. They're rules-breakingly powerful (in this case, an NPC represents getting a standard move and minor action
for free); but that's okay 'cause you can't create one, you can't control when you get one, and you don't get to prevent one from leaving. NPC allies work much the same, only they're often not in any single PCs possession.