With all sorts of CC applications, supp card applications, promos etc one can choose to use real names, or to game the system.
If gaming the system, one needs to assess the possible consequences of being found out and assessing one's own risk vs reward.
In this particular case Amex do randomly check the ID of Supp Cardholders, BUT they do not always do it. I have had various Sup Cardholders for various Amexes for over a decade and some have been checked and some not. Sometimes they have checked on application, and sometimes afterwards by separate follow up.
Many forumites have also reported similar random vetting.
Now for myself Amex has been an extremely lucrative card for points, cash, flights and other benefits. It is a 7 figure points warehouse at present as well. I certainly do push the envelope at times with them, but personally I see no point in gaming the gaining of Supp cards with false names, deliberate misspellings etc when one can easily gain Supp Cards in the names of real people.
Amex is a Golden Goose. Long may it lay such eggs.