That's a great analogy for carrier grade nat.
For regular nat it's like the pizza is able to get all the way to your house but then has no idea who to go to so somebody has to answer the door and then take the pizza from the door to the person who ordered it themselves.
And IPv6 is like the pizza delivery guy just walks right into the house up the steps into your bedroom and hands you the pizza directly.
The best part is they each have the same exact problems you'd have in real life.