Do punches and vulnerabilities have a type when using Animal Man?
Yes, their type is "Starter"

First off, the full text for Animal Man is as follows:


  +2 Power and choose a card type. Reveal the top card of your deck. If it has that card type, draw it.


The key phrase here is "card type". Page 2 of the rulebook talks about types of cards and shows that the text in the colored bar below the picture is the card type.



As we can see, both Punch and Vulnerability have a card type of "Starter". This is corroborated by the explicit listing of the card types at the bottom of the page:


  The different card types that you can play are: Starter, Villain, Location, Hero, Super Power, and Equipment. Weakness cards have no card type.