'the child' is a rather impersonal way to refer to your own child. We always referred to our son as 'the boy' when we were discussing him when we were angry about his behaviour! Also, in this context, it could be any child he was looking after, not necessarily your own. My response would be 'oh? What child is that?', assuming they had someone else's child staying with them.
Different areas and countries have different ways to express this. In the UK you'd normally say 'looking after'. Babysitting is normally used when it is not your own child.