  In this article, we present characterizations of the concavity property of
minimal $L^2$ integrals degenerating to linearity in the case of products of
analytic subsets on products of open Riemann surfaces. As applications, we
obtain characterizations of the holding of equality in optimal jets $L^2$
extension problem from products of analytic subsets to products of open Riemann
surfaces, which implies characterizations of the product versions of the
equality parts of Suita conjecture and extended Suita conjecture, and the
equality holding of a conjecture of Ohsawa for products of open Riemann
surfaces.
