My two cents as someone who’s started delving into the diversity of apple varieties:
As far as I know, there’s even less of a systematic classification for figs than there is for apples. Identifying a variety based on its characteristics is therefore probably even harder than it is with an unknown apple tree.
I’d try to find out which figs are or were commonly grown in your area, or which ones are popular. Often (at least with apples), most trees belong to a handful of popular and widespread varieties.