What units do unit vectors have?
The unit vector in the direction of v is just v divided by its lengh. But if both v and its length have units of miles, the resulting unit vector is dimensionless!
If unit vectors had units, how would one decide whether the unit vector from Boston to New York should be 1 mile long or 1 kilometer?
From this point of view, the "unit vector" one often draws in the same diagram as the original vector is really that unit vector times the constant "1 unit".