In this case, it could only be the surname, unless the father's name was Emily.
When you give your name to your child as a father, does it mean your last/family or first name? Some people inherit their grandfather's name, but not their father's, as I know.
23)Whenever I look into my daughter Emily’s face, I think about the power of fatherhood. Her face reminds me ①thatI have given her more than just my name. Like me, she has freckles stretching across the bridge of her nose almost like a BandAid. The skin around her eyes ②form small and thin lines, and the lines match mine. “When Emily smiles,” my mother tells me, “she looks just like you.” Other things in her are reflections of me. I see myself most
Customs change in different cultures, but in the UK, a child (generally) is given the same surname as her father. This is not always the case these days though. Of course, a father can choose to give his son the same first name that he himself has.