Very stupid question, but I cannot help but wonder whether measuring (and finding true distance after correcting scale distortion) of the horizontal and vertical displacement, and using the Pythagoras theorem will give the length of the path on the Mercator's map? Why/Why not.
At a local level (e.g. looking at a map of your town), using the Pythagorean theorem with the right scaling factor will be a perfectly acceptable method in practice.
However, on a larger scale like the example you've shown, there are two major issues.
(1) There is not a single consistent scale distortion on the map. The top two corners of that square are closer together than the bottom two corners are.
(2) The straight-line path from one point to another on the Earth is not a straight line on the Mercator projection, unless the two points are both on the equator or have the same longitude. Here is an image I found that shows an example of the straight-line path on a Mercator projection versus the shortest distance on the sphere:
To find the true distance between two points given their latitude and longitude, this site has a calculator, relevant formulas, and example code to compute the distance.