Absinthe is green for the same reason that leaves are green: both contain chlorophyll.
Chlorophyll is the naturally occurring green pigment in plants. Why is there chlorophyll in absinthe? Because absinthe is whole herbs distilled in alcohol, and the whole herbs contain chlorophyll which turns the absinthe green.
Over a hundred years ago, a pink absinthe was apparently sold to women, though no surviving bottles exist for purposes of verification. The pink color does not occur naturally in the herbs, so the pink absinthe may have contained dyes.
In modern day, there are also absinthes being sold with colors other than green. For example, as of the time of this writing, there is a ‘red’ absinthe on the market. Please be aware that the natural color of absinthe is green (or, in the case of some surviving pre-ban absinthes, amber brown). Any other color is not the natural color of the chlorophyll and is therefore not naturally occurring in the absinthe. Moreover, even some green absinthes derive their color from artificial dyes rather than from the chlorophyll in the herbs. Only authentic absinthes such as Lucid absinthe get their green color naturally from the chlorophyll in the herbs. Always check the ingredients list on your absinthe bottle to make sure that your absinthe does not contain any artificial dyes.