In matrimonial litigation, a minor child does not need to be a formal "party" (petitioner or respondent) for the court to pass orders regarding their welfare. Under Section 26 of the HMA the Family Court has powers to pass interim or permanent orders regarding the custody, maintenance, and education of minor children. Even if the child isn't named as a party, the court views the parents as the representatives of the child's interests. Seeking a signature for a passport is considered as interest of the child (e.g., for education or travel), and courts generally entertain such applications within an ongoing divorce suit to avoid a multiplicity of litigation.
Under the Passports Act, 1967, if one parent does not consent, the other parent often needs a court order to bypass that requirement. The Family Court, already seized of the matrimonial dispute, is considered the competent authority to decide if the refusal to sign is justified or merely an act of "matrimonial spite."
The Family Court won't "issue" the passport; it will simply pass a direction to the respondent to sign the form or authorize the Passport Officer to process the application without the other parent's signature based on the court's permission.