Children's gaming preferences, such as the types of games they enjoy, can significantly influence their engagement and learning outcomes. Research has shown that children are more likely to engage deeply and enter flow states when they play games that align with their interests. These preferences encompass specific game types, styles, difficulty levels, and social aspects that enhance the gaming experience. This study focuses on the game "Lily's Daily Life," which was chosen because research suggests that children who enjoy story-based games, particularly those involving situational narratives, are likelier to engage at an age-appropriate level. The game was designed to guide children in acquiring self-care skills through situational storytelling. This study aims to explore how children's gaming preferences interact with their engagement and learning outcomes, particularly in the context of narrative-driven games. Subsequently, qualitative research was conducted on the playing behaviors of 12 preschoolers selected from this sample, alongside a statistical analysis of flow states and achievements among 3,984 children aged 2 to 10. The results revealed that children aged 3-5 exhibited the highest flow states, with engagement significantly correlated with age (p < 0.01). Additionally, older children (ages 6-10) showed superior learning achievements compared to younger ones (p < 0.01), suggesting that age-related cognitive and developmental stages affect both engagement and learning outcomes. Qualitative observations further supported these findings, with children aged 3-5 demonstrating the most sustained engagement, while younger children required more adult assistance. This research contributes novel insights into the intersection of game-based learning, child development, and educational practice, providing valuable implications for parents, game designers, and educators in early childhood education. Furthermore, the findings offer broader implications for the integration of educational technologies, particularly in the context of ICT adoption for younger learners, emphasizing the importance of aligning learning tools with children's preferences and cognitive development stages.