Social media users and members of Prime Minister Narendra Modi's Bharatiya Janata Party have criticised the show for what they said was an incorrect portrayal of the hijackers as Hindus with Hindu names when they were Muslims.
Netflix added new disclaimers to the six-episode show last week after its officials were summoned by India's information and broadcasting ministry. It also said the code names used in the series reflected those used during the actual event.
"They have used copyright archival footage of ANI without licence, they have also used the (ANI) trademark," Sidhant Kumar, counsel for ANI, said.
"Since the series has come into such criticism, our trademark and brandname is being tarnished," Kumar said, adding that ANI wants Netflix to take down four episodes where its content has been used.
The Delhi High Court had agreed to hear the case and sought the response of Netflix, he said.
There was no immediate response from Netflix for a request for comment from Reuters, which has a minority stake in ANI.
India blames Pakistan and Pakistan-based militant groups for the December 1999 hijack, which was resolved after New Delhi freed three Islamist militants, including Masood Azhar, the head of one such group.