The Midrash (Koheles Rabah 10:16) tells us that the reason both of these women were so desperate to have the living child declared theirs was that they were both potential Yevamos (widows subject to Yibbum).
Why was the mother in law desperate to have the living child declared hers? If it’s not hers, it’s her son’s child, either way she’s exempt from Yibum. She was never a potential Yevama.