苗字にあたる部分 is just translated as "surnames" there.The book is bilingual and this is the english part: surnames of rakugo entertainers such as sanyutei,kokontei,hayashiya and katsura are called teigo.
By the way, this solves my question. Your quoted sentences so far are intermediate level, whereas your questions are the ones by beginners. You didn't answer my question in your previous thread. I'm not going to insult you, but it's because you don't, needn't, or maybe can't have your own translation, perhaps?The book is bilingual
Does it means:correspond to surnames?部分 literally means "part" here, which is the definition of the word that you'll find in any dictionary, so I suspect that it's not the meaning (or "function") of 部分 itself that's confusing you.
Rather, you're probably not understanding that 苗字にあたる is a relative clause modifying 部分 (in other words, the phrase 苗字にあたる is describing what kind of 部分 it is) and everything before など is listing up examples of that 苗字にあたる部分.
Does that help you puzzle it out at all?