Question

why unit vector does not have unit

Answer 1

unit of a unit vector is unit of the quantity of which the unit vector represents

Answer 2

Notice, that it is intuitive to assign vector $\vec{v}$ and its magnitude $|\vec{v}|$ the same dimensions.Any vector $\vec{v}$ can be seperated into two components:$\vec{v}=|\vec{v}|\hat{v}$where $|\vec{v}|$ is the magnitude and $\hat{v}$ is the direction of the vector. Therefore, by equating the dimensions of both sides we may say that $\hat{v}$ is dimensionless.