Generally, no citation styles specify that any sort of special formatting for papers with co-equal authorship. Citations exist for the purpose of making referenced papers findable to the reader, not necessarily to indicate all of the contributions of each author. Indicating co-equal authorship is generally only done in the original article's Author Notes and/or on authors' CVs.

In general, I would not try to change formatting for these types of papers, as that would almost certainly violate the style guide you are using (and would probably end up being changed by the journal's typesetters anyway).

A few questions for your supervisor and a comment:

How is one to tell if the first two authors are coequal? Must one read and interpret the (often absent) contributor-role paragraph in the article? Are you thinking only about papers with only two authors? It probably goes without saying that you should be careful how you question your supervisor. Are the cites to his or her articles? I have noticed throught my long life that when an authority figure in an academic setting makes an unusual, unorthodox demand; the danger of anger and retribution to the questioner of that demand seems to increase with the unusualness of the of the demand's deviation from the norm.

Perhaps, a couple of ways to accomplish this demand: You could make writing style changes to always mention the authors in your text and use "supress author" when entering the citation. Alternately, as the in-text citation and the bibliography entries are handled differently you can consider editing the in-text cites (_after_ you have saved a copy without Zotero codes) to add-in the second author where needed.