I suppose the number of sensory systems partly depends on how you group—or do not group—them. In addition to visual, auditory, and kinesthetic (that includes smell, taste, touch, and muscle positions, etc.), here are three other systems:
Interoception: sensors related to the condition and function of internal organs, the physical or physiological condition of our body. Thirst and hunger are examples, as well as the perception of hot or cold, or of needing to use the toilet.
Vestibular: sensors related to balance and orientation in space, including the position of our head in relation to gravity. The semicircular canals and the otoliths in portions of the ear assist in these functions. In the recent Olympic games, this was he system that appeared to malfunction when Simone Biles was competing. Gymnasts sometimes describe this as the “twisties,” where a person loses their sense of orientation when in mid-air.
Proprioception: sensors related to the position, location, orientation, and movement of body muscles and joints. Proprioception combines sensory data from neurons in the inner ear (motion and orientation), stretch receptors in muscles, and joint-supporting ligaments for stance.