First of all – this is a correct observation, as this problem does indeed exist: washing plastic dishes from fat, in contrast to glass plates, is more tedious and often requires several wash cycles. In fact, even several such washes may not suffice to rid the plastic of that 'oily' feeling, as if it's still covered by a thin layer of fat.
In order to understand why this happens one must understand how soaps (or detergents/surface active agents) work. Generally speaking, soaps remove dirt and fats by making them soluble in water.
For a material to dissolve it must share certain chemical properties with the solvent, especially with regard to its polarity (separation of electric charge): polar compounds tend to dissolve in polar solvents (for example common table salt, which is a highly polar compound composed of ions, or charged atoms, is highly soluble in water, which is also very polar). The opposite is also true: non-polar compounds dissolve in non-polar solvents (for example facial makeup, which contains mostly non-polar organic compounds that do not dissolve in water but are highly soluble in liquid paraffin, which is a mixture of non-polar compounds. So it is very easy to remove makeup using liquid paraffin). For more information on this topic please refer to "What determines the solubility of materials?".
Water and oil are very different in their polarity and are therefore insoluble in each other. Soaps are unique compounds because soap molecules contain a small polar end (known as the polar head) and a long non-polar tail:
Because of the two different parts of the molecule, a soap molecule is soluble in water and at the same time can dissolve fats. The fats are attracted to the non-polar tail part of the soap while the polar head makes the whole complex (soap + fat molecules) dissolve in water.
Cleaning plastic from fats is a much harder task for soaps, since plastics are chemically similar to fats. The main difference is that fats contain chains of about 20 carbon atoms bound to each other, while in plastics these chains are longer. Just like in the principle of solubility, similar molecules have strong attractive forces between them. This results in a sort of tug-of-war situation between the soap and the plastic, with the plastic trying to keep its hold on the fats and the soap trying to dissolve it in the water. This is why removing fats from plastic requires much harder work compared to glass or metal. Glass, which is covered with polar oxides of silicon, and the metallic surface form much weaker attractive forces with fats.
Right: the similar molecular structures of soap, fat and plastic (the soap and the plastic compete for the fat). Left: a tug of war contest in the United States Navy (from Wikipedia).
By the way, Teflon coating was developed based on the rules of chemistry described above. Teflon, or the generic name polytetrafluoroethylene, is chemically different from both water and oil. It is coated with fluorine atoms which are chemically inert, and that is why almost nothing sticks to it.
If you find these explanations insufficiently clear or if you have further questions on this topic, please write about this in our forum, and we will relate to your comments. Your suggestions and constructive criticism are always welcome.