I believe that all members here cannot answer your question because it totally depends on the context and background, which is not shown here enough.
Person A: だから、俺(person A)の補習授業にどうしても出たいって言うものだからさ、しょうがなくね。
Person B: ふむ…そうか、島津さん(Person C)だな。
Person A: 何だ、知ってるんだ。
Person B: 担任だからな…ま、補習云々に関しては小言を言うまい。私(Person B) の責務は君(Person A)を卒業させることだからな。
Person A is probably a student, who learns from person B.
Person A is a man.
Person B is a man. He is a teacher of person A's class.
Person C, 島津さん, is perhaps a woman. Maybe she is a student of that school but she is not in person A's class, or maybe she/he is not a student of that school and much older than usual students.
Person C, 島津さん attended the person A's supplimental class, I guess. Or something else that I don't know.
This is what I understand with your quote.
And your two interpretations can be possible, although it totally depends on the context.
According to this forum's rule, you should not quote any more lines here.
Therefore, unless you provide us with the precise context and background written in English, members cannot give you the correct answer.
However, in order to summerize the correct and precise context and background, you should have already known what was talking about in your quote sentences.
Then, you didn't have to ask here from the beginning because you already knew.
... I'm afraid it is not that simple question here.
If you provide the total context in somewhere else, you will get the correct answer soon enough, I guess.