The only case of Mexico using a NATO Symbol with which I am familiar is oddly on a series of pistol-caliber proof loads made by Industrias Tecnos for Spain. I have the .380 in my own collection, and I think Lew Curtis may have the 9 mm. There is a third caliber, perhaps a .38 special. Lew Curtis can probably tell us more about them.
FNM is a confusing designation, since the initials fit well with a logical name for an ammunition factory in Spanish, Portuguese, and French, at least. Also in Italian, although I don’t think they have ever used those initials with a factory name.
I don’t recall the reason for the two different FNM entries on the IAA headstamp list, for Mexico, although admittedly, I was one of the people involved with making that list. It is not up to date, still has many errors that were NOT on the original list as submitted, as badly needs someone (not me!) to upgrade it. I submitted lists of spelling errors at least twice, and they were never utilized. The box shown is a good example. Since the words used on it could be either Spanish or Portuguese, it is only the cartridge designation that reveals the country of origin.
Of course, the best thing is that you can ask about headstamps on this Forum, where you will usually get excellent answers like JonnyC’s.
As Jon ably pointed out, sometimes you have to use clues other than a headstamp to figure some of these out - primer cup material, bullet ogive, color, and presence or not, of sealants, headstamp layout (although it is not much help, generally with NATO-approved cartridges since most of them use the same trinomial headstamp), etc. Box labels help to, since they give you a language to work with, usually, and sometimes a cartridge designation specific to a particular country, as in the case of “M947” for the Portuguese box.