Microwave ovens heat up food and drinks by oscillating water molecules in the food/drink thereby increasing the temperature of these molecules, and therefore the temperature of the food/drink.
The laws of thermodynamics tell us that the higher the temperature of a material, the more thermal energy it has. And at a given temperature, the more of a particular substance, the more total thermal energy this substance will possess when it gets heated up.
In fact, the heat capacity (ability to absorb heat) is given by
$$\Delta Q = C m \Delta T$$
where $Q$ is the amount of heat that must be added to the water of mass $m$ in order to raise its temperature by $\Delta T$, and $C$ is called the specific heat capacity which is a constant for different substances. Note how the greater the mass, the greater the thermal energy that will be present.
The heat capacity for water is relatively high than compared with solids. On a molecular level, the microwaves cause the atoms of a solid to vibrate toward and away from each other like they were connected by a spring. As the temperature is increased, the frequency/energy of these vibrations increases. In a solid, this is basically the only motion available. In a liquid like water (and gases), absorbed heat energy causes the atoms in the $H_2 O$ molecule to vibrate and rotate as well as translate. This means there are more possible microstates for the water molecules and because of this their heat capacities are larger than in solids.
So if what we place in the microwave an object with more water content, the hotter the object will get compared to one with smaller water content (all other parameters being equal like size of objects, microwave power, initial temperatures of objects, cooking time etc).
Possibility one:
If we look at the left cup, we see that it is conical in shape (consider the volume per say each millimetre of height) and so there will be a greater volume of water at the top than at the bottom. This means there will be a greater mass of water absorbing energy by the microwaves at the top of the cup than at the bottom, resulting in a thermal energy gradient with a higher temperature at the top and a lower temperature at the bottom. And since the cup’s volume decreases uniformly as we move down to the bottom of the cup, so to will the temperature of the water.
We would not expect the same for the cup on the right. In fact we would expect that on average, the temperature would be the same at the top of the cup as we move down to the bottom - we also assume the the microwave transmits its radiation uniformly throughout the microwave compartment.
This seems consistent with your experimentally determined results.
Possibility two:
Microwaves also have a particular pattern of standing waves in the compartment. This could go to explaining why the conical cup had hotter water at the top than the bottom. But why would the ceramic cup have a consistent water temperature from top to bottom? A possible explanation is that microwaves are not interacting directly with the water, but with the mug itself. So the microwaves are heating the upper part of the cups directly, and even if the standing waves form at the "top regions" of the cups, there may be conduction down the length of the cup. If the cup heats up this way, then as it transfers this heat to the water, the temperature at the top would be the same as the bottom.
