Since the inception of Augmented Reality (AR) nearly three decades ago, numerous studies have demonstrated its potential to provide intuitive instructions for manual tasks within the manufacturing sector. This includes operations such as manual assembly, which can be executed with high efficiency and a focus on error-avoidance. Despite this, to the best of our knowledge, there has been no comprehensive review of the cognitive AR assembly method from a holistic perspective. This includes aspects such as context awareness, visual instructions, environment interaction, human factor considerations, and the impact on real-world AR assembly deployment. These factors are particularly relevant to current human-centric manufacturing practices in Industry 5.0. Since 2012, the release of Google Glass and advancements in artificial intelligence (AI) have significantly expanded actual AR deployments. Consequently, this review takes 2012 as the starting point for the literature collection. The objective of this article is to provide an overview of the context-aware cognitive AR (CA-CAR) assembly works published between 2012 and 2023. We aim to identify and classify the necessary context modules for CA-CAR assembly, and analyze potential technical barriers to their shop floor adaptation. It should be noted that this work offers both a historical perspective and a comprehensive map of the current research landscape surrounding the development of CA-CAR assembly applications. Furthermore, we discuss recent research trends and open problems in the field of CA-CAR assembly, along with potential future research directions.