Why can humans feel electricity?
Electrical currents stimulate neurons aspecifically. For example, the BrainPort artificial vision device conveys visual information through electrical stimulation of mechanoreceptors in the tongue. Similarly, cochlear implants and retinal implants convey acoustic and visual information through electrical stimulation of the inner ear and retina, respectively. The list goes on. Bottom line is, none of these organs (tongue, ear or eye) evolved to process electrical stimuli. And in fact, there are no specific current or voltage sensors in man. Electric currents aspecifically stimulate neurons, causing them to fire. When sensory afferents are activated in this way, sensory perceptions are generated. 

In case of lightning (electrical shock to the skin) it is mainly pain and heat receptors that mediate the sensation. In the case of the BrainPort, it is the tactile receptors on the tongue that are mainly stimulated (taste perceptions are rare). In case of cochlear implants it is auditory nerve fibers and in retinal implants it is optic nerve fibers that are stimulated. Hence, various, and in fact all, neurons can be stimulated by electric current.

Note that sharks feature Ampullae of Lorenzini. These sensory organs are electroreceptors, which allow sharks to sense the bio-electrical fields of prey under water.