It could mean that the teacher is very strict and is always on top of the person, like constantly checking on their work or behavior. For example, in a school where discipline is highly valued, the teacher might be very watchful and quick to correct any mistakes, giving the impression of being 'down his throat'.
Perhaps it implies that the teacher is highly involved in the character's development. In a positive sense, the teacher is so dedicated that they are deeply engaged with the student, always there to guide and correct. It might be like the teacher is so close in monitoring the student's progress that it feels like they are right 'down his throat' but in a helpful way.