I have always thought the word gafete meant a name tag or ID badge. I was surprised to find out that it apparently is a word specific to Mexico and there is not a real consensus as to what the word is actually referring.
Possibilities are the clip that holds the name badge on. Also it could be simply any form of ID or granted permission, though that was only mentioned once in the discussion.
Doing a google search for the word gafete is not much of a help either.
Whatever the exact definition and meaning of gafete is, it is used to mean a form of identification or temporary permission granted by wearing a badge that is easily seen.
I’m confirming your belief that it means id card. Salinas, CA has a large Mexican population and they use that term on the streets. The street vendors are now required to have identification cards and they refer to it that way.
Thanks for the comment and confirmation on the way it is used.