I'm new to SharePoint DR and wondering if the DR farm (SP 2013 on-premises) share the same Active Directory with the Production farm (SP 2013 on-premises)? Or there will be other mechanism to synchronize users between 2 separate AD? I've been searching for this specific question but did not see answer anywhere
You would usually use the same Active Directory forest/domain. Using a separate forest for the DR farm would needlessly complicate the configuration and implementation of DR.