Why refractive index has no units?

That's a good question. Basically, the refractive index is a ratio. It compares two different speeds.

It compares the speed of light in a vacuum to the speed of light in some other material, like water or glass. Both of these are speeds, so they have the same units. For example, metres per second.

When you make a calculation and you divide one speed by another speed, the units cancel each other out. The 'metres per second' on top gets cancelled by the 'metres per second' on the bottom.

So, you are just left with a number. This number tells you how many times slower light is in that material. It’s just a pure comparison, so it doesn't need a unit.