When you shake absinthe in a mixing tin with ice, the oils from the absinthe start to separate out of the elixir. This phenomenon is not immediately apparent. Rather it takes a few moments of shaking to really see it happen. What does that mean? It means that the more you shake the more you change the characteristic of the base liquor. It means that you can literally shake the flavor out of many brands of absinthe. Not all of the flavor perhaps, but a noticeable amount.
So if you want to make a drink and you are finding that the anise is nice but it is overpowering the other flavors in the mix you may want to just shake it a bit longer. This adds a bit more water to the mix but it also means that more of the oils will separate out of the drink and stick to the side of the glass.
This can also change the look and feel of the drink. For instance it can make a drink seem oily or it can leave a film floating on top of the drink which may be undesirable. This usually takes considerable more shaking than one would do when making a cocktail.
Currently there is only one brand of absinthe available on the US market today that does not break down when shaken with ice, that brand is Le Tourment Vert. This is not surprising since Le Tourment Vert is the first real American style, cocktail absinthe. From the beginning it was really formulated to be mixed in cocktails. That is very different from traditional brands which have a flavor profile that is more appropriate to a traditional presentation with water and perhaps a bit of sugar.
In this video Mixologist Jared Williams demonstrates this phenomenon.