The high moisture extrusion of plant proteins is well suited for the production of protein-rich products that imitate meat in their structure and texture. The desired anisotropic product structure of these meat analogues is achieved by extrusion at high moisture content (>40%) and elevated temperatures (>100 °C); a cooling die prevents expansion of the matrix and facilitates the formation of the anisotropic structure. Although there are many studies focusing on this process, the mechanisms behind the structure formation still remain largely unknown. Ongoing discussions are based on two very different hypotheses: structure formation due to alignment and stabilization of proteins at the molecular level vs. structure formation due to morphology development in multiphase systems. The aim of this paper is, therefore, to investigate the mechanism responsible for the formation of anisotropic structures during the high moisture extrusion of plant proteins. A model protein, soy protein isolate, is extruded at high moisture content and the changes in protein–protein interactions and microstructure are investigated. Anisotropic structures are achieved under the given conditions and are influenced by the material temperature (between 124 and 135 °C). Extrusion processing has a negligible effect on protein–protein interactions, suggesting that an alignment of protein molecules is not required for the structure formation. Instead, the extrudates show a distinct multiphase system. This system consists of a water-rich, dispersed phase surrounded by a water-poor, i.e., protein-rich, continuous phase. These findings could be helpful in the future process and product design of novel plant-based meat analogues.