Question

refractive index has no units why?

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.

Answer 2

"Refractive index" is a dimensionless unit. This is because the "refractive index" of any object is the "ratio" of "speed of light" in vacuum and "speed of light" in given medium.

Hence it can be written as:

$\Bold{ \text {Refractive index}=\frac{\text { Speed of light in vacuum }\left(\frac{m}{s e c}\right)}{\text { Speed of light in given medium }\left(\frac{m}{s e c}\right)}}$

Both units of speed will cancel with each other.  That makes refractive index as a unitless value.