My opinion: Chai (tea) is nothing but rechristening of the Indian chai for the export market under the name of chai teachai tea.
- The "chai tea" usage would be understood but frowned upon in India.
Similar usage-
Numerous United States coffee houses use the term chai latte or chai tea latte for their version to indicate that the steamed milk of a regular latte is mixed with a spiced tea concentrate instead of espresso. (wiki)