The rapid advancements in foundation models and sixth-generation (6G) wireless communication systems necessitate the development of efficient, scalable, and privacy-preserving machine learning approaches. For foundation models in 6G, split learning (SL) and aggregation learning (AL) have emerged as promising paradigms that address key challenges in distributed artificial intelligence (AI), such as communication efficiency, resource allocation, and data privacy. SL enables multiple entities to collaboratively train deep learning models by partitioning neural networks, while AL focuses on aggregating intermediate results or model updates from multiple participants, improving robustness, optimizing resource utilization, and mitigating data leakage risks. Specifically, SL is ideal for scenarios requiring strict data isolation (e.g., vertical collaborations), whereas AL suits homogeneous horizontal data settings; they can be combined to balance privacy and communication efficiency. This survey provides a comprehensive analysis of SL and AL in 6G communication systems, exploring their architectures, technical methodologies, and integration with AI-native 6G communication technologies. We examine different SL configurations, aggregation techniques, and their roles in optimizing distributed foundation models. Furthermore, we discuss their applications in emerging wireless networks, including semantic communication, reconfigurable intelligent surfaces (RIS), space-air-ground integrated networks (SAGINs), and quantum communication. By analyzing the impact of SL and AL, this survey provides insights into their role in shaping distributed AI-driven communication systems in the 6G era, focusing on efficiency, privacy preservation, and scalability.