What? Where do you live? Third world country? Does your country still allow children to have less than 2 parents on their birth certificate?
I think you read too much into it. Consider the context. I am talking about the what-if "me". When I talk about myself and branch off to talking about my kid, it's usually going to be "my kid" or "my son" (as here).
If I'm in a discussion where I'm talking about the unit "us" as parents, it is "our kid" or "our son", and it's the same if my wife and I are discussing our son. (Seriously - that 'our son' wasn't contrived, it's all about the context. I'm talking here about both of us, so I automatically wrote "our son")
And when the kid in question pisses me off, it's definitely "do you know what your son did?" ;)
They're her kids (as far as we know) at least as much as they are his.
Also, the story used the word "girlfriend", not "wife". They're not married, so he has no legal claim to them.