It arises from the investment of devotees- from their ability to imaginatively interpret certain canonical works
Nowhere is this better illustrated than in the emergence of 108 Divya-Desam canon in SriVaishnava tradition
Sri-Vaishnavas have this list of 108 Divya-desams - Vaishnavite temples concentrated in Tamil country
These are "Divya Desams" as they have been "eulogized" by Alwar saints in nAlAyira Divyaprabandham - 4000 Tamil verses from 6-9th cen
E.g. The kRSNa temple at Mannargudi, which is an "Abhimana sthalam" but not a Divya desam.
DId the Alwars indeed mention these 108 places in their verses?
Actually not necessarily
It arises from the ways in which commentators have interpeted the verses centuries later!
Periyavachan Pillai (who wrote a commentary on each of the 4000 verses)
Pillai Perumal Iyengar (who used Periyavachan's commentary to create the list of 108 DDs)
Pillai Perumal Iyengar then leveraged Periyavachan's commentary to create a list of 108 pilgrimage centers, which were spotted by Periyavachan
Kumbakonam's Aravamudhan being one of them
Hence it becomes a Divya Desam
Did Nammalvar intend it that way? Not sure. But the association with the place happens in the writings of the two men I mentioned