It is not an error to situate Khayyam in medieval Muslim culture.
Khayyam’s Persia heritage is relevant, as Iran contributed mightily to the Islamic Golden Age.
It would be an error, however, to suggest that the rest of the Muslim world languished benighted while Persia alone shone. Spain gleamed as brightly; Mesopotamia & Egypt, nearly so.
In every case, local talent achieved greatness through interaction with the wider Muslim world, and with nearby non-Muslim cultures. No civilization excels in isolation.
The same logic holds true for the European Renaissance, too. Da Vinci’s greatness owed less to his native language and culture than it did to his region’s close connections to the Muslim and Byzantine worlds, and to the rest of Europe, as the headquarters of the Roman Catholic Church.