What is PostGIS doing with ST_DistanceSphere?
There are two problems here.


Though it calculates the distance on a sphere, PostGIS doesn't calculate with the major-axis radius in WGS84 as presumed in the question. The sphere is created with a radius that is the mean radius of the WGS84 ellipsoid. This is by design.
Neither does MySQL, however the radius in MySQL was lifted from a lazy-copy-job from PostGIS that converted a radius in miles to meters from an obscure constant from a random 20-year old PostgreSQL module. Far from by design, this is presumed to be a bug.