something to keep in mind is the history of the spread of Islam through the expansion of the Ottoman and Safavid empires from early on in the middle ages until the 20th century. Especially notable is the tolerance of the Ottomans for the practices of other cultures which helped in it's flourishing for almost 700 years with holds as far north west as Budapest and down south to North Africa, and the shores of the Red Sea.

This allowed for local practices to persist even under Islamic rule, and perhaps somewhere along the way, FGM began to become associated with Islam since it was being practiced under Islamic rule.

This is also the way veiling came to be analogous with Islam. This is completely unrelated but just goes to show the way cultural and religious distinctions become blurred over time.