Legally speaking, he can’t claim her as a “qualifying child“ dependent because she does not live in his home more than half the year. He can’t claim her as a “qualifying relative“ dependent, because, regardless of the other tests for that type of dependent, a child can’t be claimed as a qualifying relative dependent by one taxpayer if they can already be claimed as a qualifying child dependent by another taxpayer. Both you and the child’s mother seem to live with the child more than half the year, which means that either you, or the mother can claim the child as a qualifying child dependent. That means that the father can’t claim the child, even if you were willing to allow it. (He can’t claim the child because the child can be claimed by you, so even if the child is not actually claimed by you, he is still disqualified.)
