We cannot pin down the linguistic identity of the early medieval Acharyas
Adi Shankara never wrote a word in the vernacular. Nor did Ramanuja. Their writings are entirely in Sanskrit
Malayalam as a distinct language likely did not exist in 7th cen CE
They were cosmopolitan men who wielded a cosmopolitan language arguably not just in their work but also in day to day speech
Quite likely many of these acharyas hailed from families that were recent migrants from the north
That's not the Indian way...In India groups retain their distinctive identities long after migrating to a new region