There are 2 reasons.
Number 1 - Because Bengal is called Gauda Pradesh therefore the Sampradaya of Gauda is Gaudiya Vaishnava Sampradaya. North is Gauda and south in Dravida.
Number 2- Gaudiya Vaishnava Sampradaya is a lineage of the Madhva Sampradaya, our Acharyas have explained that one name of Sripad Madhvacharya is Gaudaprajna therefore the Word Gaudiya ...
Gaudiya indicates that they honour Radharani along with Sri Krishna
A Krishna centric devotee is technically called a ”Karshnav”
Someone who worships Radharani along with Krishna,is called a Gaudiya Vaishnav.
This is explained in a lecture by HG Amogh Lila Prabhu,6:10-6:49
From this blog which answers Who are Gaudiya Vaishnavas?
“Gauḍīya” indicates the part of India between the southern side of the
Himalayan Mountains and the northern part of the Vindhyā Hills, which
is called Āryāvarta, or the Land of the Āryans. This portion of India
is divided into five parts or provinces (Pañca-gauḍadeśa): Sārasvata
(Kashmir and Punjab),...
Nimai took Sannyas from Keshav Bharati in Katwa (West Bengal). After the initiation, Keshav Bharati gave Nimai the new name "Sri Krishna Chaitanya." It literally means Krishna consciousness.
Chaitanya Mahaprabhu: the Story of Bengal's Greatest Bhakti Saint by Chitrita Banerjee
Don't know about that story but here are some points from scripture that are used to justify superiority -
As stated on this webpage - Apart from the Padma Purana, does any other Purana say that Adi Shankaracharya will teach mayavadi philosophy? - There are puranas which talk about Shankara spreading false doctrine.
Vedanta is rejected by Shaivas and ...