Ты любишь свою фамилию. -- You like your (own) surname.
Он любит свою фамилию. -- He likes his (own) surname.
Она любит свою фамилию. -- She likes her (own) surname.
свою means as much as "one's own" and refers to the subject of the sentence. In Russian it's always свою, in English it's translated to his/her/your depending on who is the subject of the sentence.
Она = She, so it has to be "her" (own) surname.
If you wanted to say "She likes your surname", it would have to be Она любит твою фамилию.
Friend detailaddict: Russians generally have three names. The “first” name is "имя". The second name, or “patronymic”, "отчество" in Russian, is derived from the person’s father’s name. The “last” name, or surname, in Russian is the "фамилия". Most Russian adults, I’ve read, address each other by their first name and patronymic. And so, I think the answer to your question is "no".