TL;DR: it is due to the effect of a mechanical edge.
When you are holding a cup pressure is exerted by the fingers (essentially). Like the following image. Pressure is always normal to the surface.
That pressure results in a force (black arrow in image below) that if it is analysed in the vertical and the horizontal you get the following.
you will notice that the vertical component is opposite in those cases. I.e. when you increase the pressure the cup will be pushed downwards.
One thing that pop to mind (excuse the pun) wat the ice lollies (a type of frozen icecream in a cone shaped packaging) . If you ever held one of those and tried eat it, you know that you need to squeeze it, to make it pop out.
The principle is the same. If you inverted the cone then as you increased the pressure the glass would be pushed downwards (more easily that in the opposite case).
Final note: Lab glassware
One of the few exceptions, when an inverted cone shape is used is in Lab glassware like the following.
However you may notice that there is still a part of the glassware with a straight neck and also there is a visible bottle lip at the top (for lab clampers to be able to hold). The main reason that this the preferred shape, is that it secures the liquid in place and it enables easier stirring with little chance of spilling.
Why would they be easier to hold?... perform an experiment ... decide for yourself $\endgroup$