In English we can say
His job was to ...
and we can say
It was his job to ...
There's hardly any difference in meaning, but the first tends to suggest that this was all of his job: that's what he did, and it might almost define him (in the way we do define people by their jobs: she is a doctor, he is a teacher).
The second suggests that this is not all of his job, but just what his job is in the particular context we are talking about. It is less likely to define him.
As I said, the difference is subtle, and these suggestions can be contradicted by other things said.
So, in the given example, whose job it was corresponds to it was his job; whereas whose job was would correspond to His job was. The use of it was suggests that teaching him how to be a prince was not the whole of the tutor's job.
You asked in a comment "Why isn't it written this way? Whose job it was to teach that how to behave as a true prince."
First, that wouldn't be grammatical: you can't use that as a pronoun referring to a human. It could be to teach him how to behave as a true prince.
The phrase used is to see that he knew how to behave. This use of see (sense 5 here) means that "ensure". So "see that he knew" is very similar to "teach", but implies something less formal than teaching.