I believe the prayer rug comes from the Ottomans (uncited) who added prestige to the artifacts. After the conquest of Constantinople, the Ottomons adopted the Roman (Byzantine) Orthodox Church's symbol () to represent Islam (Crescent and Star). They adopted the church dome and bell tower for architecture of the mosque called dome and minaret. And now it seems they also adopted the prayer rug.

(Fun anecdote: I traveled to Venice and the tour guide said the architecture of the St. Marks Basilica is different from the Catholic Churches of the rest of Italy as it resembles Islamic Architecture influences. Ha! It resembles Byzantine or Constantinople influences, not Islamic at all but quite similar.)