I suspect this refers to his degree – Master of Arts (Artium Magister), now more commonly abbreviated as M.A.
In the vital records you found this entry in, there is another John Rogers (d.1782), a Reverend (and apparently the father of your John Rogers), who also has A.M. after his name.
I could not find an image of the headstone for John Rogers d. 1827 – ...
What I do know about Argentina is that they keep your payment and provide nothing!
They happily answered my emails, and advised me how to apply for a birth cert. So I did, and paid, then no certificate and a wall of silence ever since.