I submitted a pure mathematics paper (my first paper!) to a journal in September 2020. It is now December 2021 and I have received zero reports. A few months ago I sent an enquiry. I got a response saying that I should hear back soon and that there is a very large backlog due to administrative issues. My thesis adviser told me that this is extremely unusual (I'm not sending it to the Annals). What is the correct protocol for how one should proceed?
A few months ago I sent an enquiry. I got a response saying that I should hear back soon and that there is a very large backlog of administrative issues.
You did hear back. You just did not get a review report.
What to do is a difficult question (see e.g. this question), but in your case there's an easy answer. Ask your thesis advisor and do whatever they say - they are after all much more experienced than you, know the contents of your paper, know the review process of your field, and know what tier of journals to aim for. If they're an author on the manuscript, you'll also need their approval to submit elsewhere, per standard protocol.
They have responded, and there is not much you can do to force their hand, especially if they have "a backlog of papers" to choose from. You likely have expedited your process by messaging them already. It could have been lost in the shuffle, assigned to reviewers who are tardy, or the reviewers could be hard to find for your sub-area.
If this paper is particularly important, you may be able to inform them of why. Editors have ways of expediting processes for those who need it. You could potentially skip the queue or get assigned fast and trustworthy reviewers. For example: "I am up for (tenure review)/(dissertation defense) on XYZ. Is it possible to get an estimate of about how long it will take for a response?"
If you are particularly in a rush, you can even threaten to withdraw the paper. Only do this if you really would do so and really do need a response soon. Some sample text might be: "This timing is extremely important to me, and I may need to withdraw if I cannot have a response by then."
However, this is your first paper - don't rush it. You want a response now, but don't need it. If this journal was your top pick, the best end destination for this article - do not withdraw because you are impatient. You have plenty of time to rush articles later. Start your career off right with the best publication(s) you can. The months it takes to get a response will be worth it.