Cyclists often complain of pain, tension and burning in the shoulder/neck area or in the arms. The main cause of this is the aerodynamically stretched riding position. Added to this is often a pathologically altered spine, such as displaced intervertebral discs or worn vertebral joints. This can narrow the nerve root holes through which the nerves exit the spine. The pinched nerves react irritably.
If you also hold your head in a reclined position for a long time - like a racing cyclist - the nerve root holes become even narrower. This can lead to the sensory disturbances and pain you mentioned and to a vicious circle of nerve irritation, tension, blockage and further irritation.
However, you can break the cycle: If the discomfort only occurs on the bike, change your position so that you are sitting as upright as possible. The shorter the stem and the higher the handlebars, the better. You should also frequently change your grip on the handlebars while riding to change your sitting position. Wear clothing that protects you from draughts. Exercise your neck and shoulder muscles and stretch and relax tight muscles. If none of this helps, you should talk to your doctor about methods such as osteopathy, acupuncture or neural therapy.