This trope is related to the Papa Wolf trope, but while most examples of this trope are a Papa Wolf (or the occasional Mama Bear), not every Papa Wolf falls under this trope as it differentiates itself in a number of ways:
Badass and Child Duo requires The Badass to have been a Badass from the very beginning, and at all times, but Papa Wolf can be any kind of person weak or strong who will become badass when their children are threatened.
In Badass and Child Duo, The Badass is responsible for one child, usually not related to them in anyway, while the Papa Wolf's children are usually their own kids and/or their kid's friends.
The Papa Wolf becomes badass usually for a specific situation, and may be a one-off thing, while Badass and Child Duo actually requires that the child constantly be in danger for more or less the entire work so as to invoke the Badasses skills.
Papa Wolf has no personality or age requirements or common attributes, while Badass and Child Duo does for both the Protector and Protectee, in particular the child must actually be a child or early adolescent, while the Papa Wolf's children can be of any age, even adults.