In the context of neuromorphic computing chip engineering, this review paper explores the area of bio-inspired artificial synapses with a focus on the incorporation of soft biomaterials. Soft biomaterials, including biocompatible hydrogels and organic polymers, have definite advantages in resembling the soft and dynamic properties of biological synapses. The article gives a general review of neuromorphic computing while emphasizing the shortcomings of traditional von Neumann architectures in terms of emulating the functions of the brain in computing. It highlights the artificial synaptic design concepts, including synaptic plasticity and energy efficiency. Spike-timing-dependent plasticity, synaptic weight modulation, and low-power operation can all be incorporated into these synapses thanks to the use of soft biomaterials. Inkjet printing, self-assembly methods, and electrochemical deposition are only a few of the technical techniques covered in this article for creating artificial synapses that are inspired by biological structures. These methods enable accurate biomaterial patterning and deposition, enabling the construction of complex neural networks on neuromorphic circuits. The research also emphasizes possible uses of bio-inspired artificial synapses in robotics, prosthetics, and cognitive computing. Soft biomaterials' capacity to mimic the synaptic activity of the brain creates new opportunities for effective and clever computing systems. In summary, this review paper succinctly outlines the incorporation of soft biomaterials into artificial synapses that are inspired by biological structures for neuromorphic computing chip fabrication. It analyzes production methods, highlights the value of synaptic plasticity and energy efficiency, and examines prospective applications. The development of new computing paradigms and the creation of extremely effective and brain-like computer systems are both significantly impacted by this research.