if you are watching directly from the above (incident angle is 0°) then the Real depth = Apparent depth
To form the final image the objective of the travelling microscope collects a cone of rays with a range of small angles of incidence (as assumed in the derivation of the formula for refractive index) from a point on the object and not just one ray with an incident angle of zero degrees.
Update as a result of a comment from @SivaManasan
The derivation of the formula for the refractive index is done in many books and in many Internet sites but they all make an approximation of the type if angle $\theta$ is small $\sin \theta \approx \theta,\: \sin \theta \approx \tan \theta $ etc and here is one of those derivations.
My point about the cone of rays I have tried to illustrate below.
The first thing you need to know is that you only move the travelling microscope you make no adjustments to it so the distance between the object and the final image is fixed.
These are distances $CC'$ and $DD'$ in my diagram.
The other important property of the microscope is that its depth of field (distance between the nearest and the furthest objects that are in acceptably sharp focus in an image) is small.
Without the glass block (grey rays) the cone of rays $ACB$ originating from a point object are focussed by the microscope lens $AB$ to form an image in the plane of the cross wires of the microscope at $C'$.
Putting the glass block in the way effectively moves the observed position of the object for $C$ to $D$ and now $D$ acts as the object for the microscope lens.
To focus the cone of rays apparently originating from point $D$ the microscope lens has to be moved to position $A'B'$ which is distance $h$ in my diagram.
On moving the microscope (lens) the cone of rays $A'DC'$ are now focused by the lens at $D'$.
Note that all the rays (except the zero incidence ray) undergo refraction (bending of a light ray as it travels from one medium to another) and so the derived formula for the refractive index is valid particularly as the angles involved are small because of the small diameter of the microscope lens.
Most of the resources I've seen are from Indian sites. Do western
countries use different method or instrument?
There are other methods with pins etc but one of the main reasons why Indian sites have a lot of reference to to this method of measuring refractive index might be that the topic is in the syllabuses of Indian examination boards whereas, for example in the UK, this method has been omitted in favour of other methods which mostly rely on the measurement of angles.