The Queen (or King) is not the government; she represents the State. The difference is often ignored by ministers, but is important particularly in constitutional cases.
Miller began as Miller v Home Office, a judicial review case. When it became clear that the question was what powers the government actually had in a certain situation, the Supreme Court decided that constitutional points should be argued by, effectively, an amicus curiae on behalf of the state, with government lawyers defending their own viewpoint (and other interested parties intervening). This made it, in their view, a case of the state versus the government, with 'Miller' being either an acknowledgement that the applicant remained a party or a means of distinguishing this case from all the other "R. -v- Government" cases over the years, depending on your point of view.