Can I add my child to W-2 if my partner also has added her to his W-2 and also can i claim her on my taxes since we are not joint filing and not married.

My partner does have dependent care set up through his work, but I have paid day care out of pocket and other expenses. He told me that I am not allowed to claim her on my W-2 because he is and on my taxes, but I do not find that to be fair. We are not married and do not share finances so there has to be a way to claim for me also.