For those who care about historically accurate takes, he only self-identified as a Muslim from Khwarazm. Self-identifying like that was the norm.
To call him Persian is a typical western post-christian modern take with its over-concern with everyone's supposed ethnicity/races.
He came from a region where Turkic people lived under the direct rule of the east-Persian Sunni Samanid amirate (under Abbasid suzerainty). So he probably was of some Turkic background, but we don't actually know. People at the time could move around freely, and many of them did.