Question

why the refractive index has no unit

Answer 1

Because the refractive index is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium .

As it is a ratio it is unitless .

The units m/s or km/h cancel out each other.

HENCE it has no units

Hope it helps

Answer 2

Refractive indices (singular: refractive index) have no units. This is because, as is evident, that they are ratios of speeds of light in a given pair of media.
Refractive index⇒ c/v
                          ⇒(m/s⁻¹)÷(m/s⁻¹)                {everything cancels out}
   where :                          c=speed of light in vacuum
                                         v= velocity of light in the given media
                         
 Another way to look at it is that it refractive indices are constant for a given pair of media (for eg. air and glass). And constants are dimensionless on their own.