that is not the case. if you have an icici-jet cc, it does not affect your ability to have an amex-jet cc. all they are saying is all your other amex credit cards will be cancelled. amex rules provide for one charge card and one credit card per person. they do make exceptions, however, rarely!
That is not true in the US - we can have an SPG-AmEx, a Gold Amex, a Plat AmEx and may other variants!