Short Answer: Though the concept of development of perception in children is known (Vernon et al, 1961), the process of differentiation between different senses is largely innate, though some disorders related to sensory processing may be acquired during infancy.
Background: The processes involved here are sensory processing and sensory integration. Sensory processing is a process that organizes sensation from one’s own body and the environment, thus making it possible to use the body effectively within the environment (Stein et al, 2009). It deals with how the brain processes multiple sensory modality inputs, like proprioception, vestibular system, auditory system, vision, etc. into usable functional outputs.
The second process is known as sensory integration. Sensory integration is the process through which the brain organizes different sensory informations. In short, it is the organization of different sensations for use by the brain (Ayres et al, 2005). The theory of sensory integration was put forward by Anna Jean Ayres.
As it seems obvious now, sensory integration is an innate process (Cronin et al, 2015). However, sensory integration disorders can be acquired by a child in the early stages of life. This can cause results similar to what you say. For example, according to a 2009 study, every 1 in 6 children has sensory issues which make it difficult for them to process different sensory stimuli. In fact, some symptoms which are common for children with sensory processing disorders are4:
They can’t sit still through a half-hour lesson and disrupt the class.
They often seem distracted and don’t pay attention to what teacher is saying.
They bump into kids in the lunch line, making them angry.
They can’t hold a pencil correctly, so they struggle with handwriting.
They get upset when asked to switch from one activity to another.
They melt down during assemblies and have to leave the gym.
Such activities are often seen as indications of problems with sensory processing and sensory integration i.e. they cannot differentiate between different senses or have problem organizing these sensory inputs to get meaningful output. However, I didn't find a case study where a person is unable to tell whether the input stimulus is touch or taste. This might be because (mostly) different senses are processed in different areas of the brain (Pirotte et al, 2008).
References:
M. D. Vernon, The development of perception in children, Educational Research 3 (1961),
no. 1, 2–11.
Stein BE, Stanford TR, Rowland BA (December 2009). "The neural basis of multisensory integration in the midbrain: its organization and maturation". Hear. Res. 258 (1-2): 4–15. doi:10.1016/j.heares.2009.03.012
Ayres, A. Jean (2005). Sensory integration and the child : understanding hidden sensory challenges (25th anniversary ed., rev. and updated / by Pediatric Therapy Network ; photographs by Shay McAtee. ed.). Los Angeles, CA: WPS. ISBN 978-087424-437-3.
A. Cronin and M.B. Mandich, Human development and performance throughout the lifespan,
Cengage Learning, 2015.
How Sensory Processing Issues Affect Kids in School - Child Mind Institute
Pirotte B, Voordecker P, Neugroschl C, et al. (June 2008). "Combination of functional magnetic resonance imaging-guided neuronavigation and intraoperative cortical brain mapping improves targeting of motor cortex stimulation in neuropathic pain". Neurosurgery. 62 (6 Suppl 3): 941–56. doi:10.1227/01.neu.0000333762.38500.ac