No, they ask this "are you an existing Axis Bank customer?" thing for every card applications. It's a regular thing.
But like
@Advik bro mentioned above, this card has lower approval rates. Unless you are an existing Axis Bank card holder with good usage patterns and timely repayments, most likely the card will be rejected.
In some rare cases, if someone is having good credit scores, this card may get approved without having an existing relationship with Axis Bank.
This is the most valuable co-branded credit card that's why it goes through all these verifications.