When I go into dissection rooms at medical school the cadavers do look different in terms of colour but this is because the blood has more or less been totally drained, so the skin has lost its characteristic pink colour and looks more waxy. Obviously the skin tiself does change in texture but the cadavers I see have been that way for weeks, so at funerals you would not see this. The skin just looks paler than normal.

The embalming fluids used in funeral homes to prepare bodies for viewing contain coloring agents to impart a (hopefully) natural color to the skin. In life, the blood in the capillaries gives color to the skin, even in dark-skinned persons. The embalming fluid replaces the blood, so a coloring agent is helpful in maintaining appropriate coloring to the skin. The embalmer can add additional coloring to the embalming solution as needed, such as to overcome the yellow skin tone of jaundice. The more color that can be added internally through the embalming fluid injected, the less external cosmetics need to be applied to achieve a suitable appearance for viewing.

Both of these answers are excellent examples of the different ways a deceased person can look. The embalming fluid itself is a mixture of chemicals that by themselves do not add any color to the skin of the deceased. In many funeral homes, a red or pink dye is added to the mixture so as to give a more natural color to that of the person who has passed. It's really only used in Australia for the purpose of a "viewing" prior to a funeral these days. The open-casket funeral has long been replaced and preference has been given to a viewing prior and a closed casket during the service. These days, Formaldehyde is the most common of the embalming chemicals used, but as this is a very dangerous chemical, there are new, more natural embalming fluids being developed, one of which my own funeral home uses, and has proven to be much nicer in giving a very natural, sleep-like look to the deceased. It is completely non-toxic and therefore is preferred by many over the traditional embalming fluids. But in answer to your question, no, the embalming fluids themselves do not change the color or the deceased, it is only because of a dye added to the mixture that will give this change, or the use of the more natural embalming fluid that already has the dye in the mixture.

It used to, but nowadays they use food coloring in it (red, of course) so that the body doesn't look so "dead". It is purely for cosmetic purposes only, so that the famioly can have an easier time of things.