Researchers across domains have investigated children's use of LLM-based chatbots through various perspectives and methodologies. However, prior research remains fragmented regarding anthropomorphism, the tendency for children to assign human characteristics to those large language Model (LLM) chatbots as non-human objects. By analyzing 35 empirical studies published between 2022 and 2025, this systematic literature review identifies the drivers of anthropomorphism in children's LLM chatbot interactions and the subsequent outcomes of these interactions. We found that human-like persona construction, adaptive scaffolding, supportive companionship, and non-human embodied design drive children's anthropomorphic interactions. Additionally, five anthropomorphic outcomes emerged, including children exhibiting paradoxical social and moral responses, dual consciousness about the chatbots, forming varying social ties, exploring social boundaries, and attributing human narratives to conversation breakdowns. The findings, including both benefits and risks, can inform the future design and development of LLM chatbots focused on children's well-being and promoting sustainable interactions that meet children's developmental needs.