The presence of social robots in children's daily environments has steadily increased. With the advancement of artificial intelligence (AI), social robots have influenced children's learning and development. This study innovatively utilized the Web of Science database and conducted a bibliometric analysis of 517 publications on social robots supporting children's learning and development before September 2022. Unlike most existing reviews, this study employed a synergistic combination of two complementary visualization tools, VOSviewer and CiteSpace, to map the intellectual structure and analyze the knowledge evolution path in this emerging interdisciplinary field. Specifically, VOSviewer generated visualizations depicting collaboration networks, research hotspots, and trends based on co-occurrences. CiteSpace enabled quantitative measurements of node centrality and burstness to reveal pivotal entities and emerging topics. Combining visual mapping and quantitative analysis by VOSviewer and CiteSpace allowed comprehensive landscape mapping for an in-depth investigation into the development of this field. This study proposes future research directions, including children's perceptions of social robots, social robots enhancing children's learning, social robots supporting children's social and emotional development, and social robots for children with special needs. The findings also inform the design and application of child-friendly social robots equipped with generative AI techniques.