The answer to this question is, “No”. While studying in Australia, it is a visa requirement that you hold an OSHC policy. Whilst most educational institutions sell OSHC to their students, OSHC itself is not tied to your educational provider.
Some educational providers have preferred provider status which may provide limited extra value to students, which may be a reason to change OSHC providers when you are changing educational institutions. If you are going to change, you should first compare OSHC providers to ensure that you are getting the right cover at the right price.