Baby last name

So I gave my baby just the father’s last name. & my mother comes at me mad bc I didn’t give my child my last name. She said the baby’s father could take him from me (if it ever came to that) since it’s just his last name. So now I have to give him his and my last name. Is this true? I live in Maryland.

So now I gotta call insurance & tell them I’m adding my last name & I already received a paper insurance card thingy.