Why are plastics lipophilic? My wife prepares lunch for me every morning. After eating it on my desk, I typically wash the lunchbox and utensils in the company's kitchen.
I have noticed that it is very difficult to remove the oiliness from the plastic containers (think Tupperware, but also harder plastics that look like polycarbonate): I need to rub the detergent-soaked sponge extensively and rinse with very hot water, several times. This does not happen with glasses, porcelain dishes or metal utensils like forks. 

In other words, all the plastic containers I have come in contact with seem to be quite lipophilic.

Is this because edible oils and plastic are chemically "similar" (i.e., both contain long-ish chains of hydrocarbons), or is there another reason that I am missing?
First, many plastics are somewhat porous. Since plastics are permeable to water vapor and oxygen, coffee and other food products are sometimes packed in two-layer bags of polyethylene (for strength) laminated to aluminum foil (to reduce permeability).

Second, lipids and other non-polar substances can dissolve in and and migrate through common plastics used in food containers, such as polyethylene and polypropylene, as you and @Alchimista note. This is why carotene, lycopene and other chemicals in tomatoes eventually stain poly containers permanently: they've dissolved into the plastic.

In a similar way, mercury dissolves into gold (e.g. jewelry) in a matter of seconds, leaving a silvery stain. That stain then disappears in a week or two as the mercury migrates throughout the object.

The reverse phenomena also occurs, where substances in plastic packaging can migrate out of the packaging.

As the Bard might have put it, "Will all the water in the ocean wash this food from my cans?"