He spoke Arabic and written in Arabic too. Doesn’t make him an Arab (It does actually if it was his native tongue, unlike Persian where to the best of my knowledge, merely speaking the language is not enough).

Arabic and Persian were the languages of science and the court under the Samanids. Some of that originally-Samanid Persian+Sunni influence actually persists until today. Persian is for example still today a primary, if not the primary, language in Religious schools/madrasas in the Indian subcontinent. Doesn’t make the people there Persian.

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.