The video game Animal Crossing: New Horizons (AC:NH) launched during the COVID-19 pandemic and families turned to it as a game to play together during isolation. This interview study of 27 families considered how families used AC:NH for Joint Media Engagement (JME), where family members engage with media content together, interacting with each other and bringing additional meaning to the experience. We find that the design of AC:NH well facilitates Takeuchi and Stevens's six conditions for productive JME. Furthermore, we identify and discuss additional conditions that contribute to productive JME: variety and flexibility in play styles that amplify mutual engagement, support for disentrained play that enables new forms of "joint" engagement, and scaffolding for affective interactions. This is followed by an exploration of how the COVID-19 pandemic affected JME. We conclude with design implications for building games to support productive JME for families through design for persistent shared spaces, flexible in-game progress, and social life simulation.