There are two things here.
One is that the job TITLE does not matter at all. An employer can call a job anything he likes, that makes sense within his business. It doesn't have to mean anything to an outsider, e.g. 'Supervisor, Department C' or 'Assistant to Mr Johnson' could be anything at all, till you find out what Department C, or Mr Johnson, is occupied with. So when finding an ANZSCO code under which to apply, it is necessary to find a DESCRIPTION that matches your work, rather than matching the names given by your boss, and ANZSCO.
INZ sometimes change the ANZSCO code, if they can see that an applicant has made a slightly wrong choice, and a different code would actually be a better fit with their work as described by themself and their employer.
So the second thing is, if the job description from ANZSCO is a good match with your job description, that is all good, and the names/headings don't matter at all. If, on the other hand, there seems just to have been a mistake, and the ANZSCO description being quoted is nothing like, or not much like, your job, then you need to contact your CO and point this out.