Basically a base chemical
Vodka is the simplest, the most utterly unsophisticated alcoholic beverage of them all -- consisting of essentially nothing else than a mixture of 40 % ethanol (C2H5OH) and 60 % water (H2O). Of course, there may be minute amounts of impurities in the brew, but they are truly minute, mind you. Still, for the sharp-sensed connoisseur these impurities can conceivably serve as distinguishing marks for different brands of vodka (though I have my doubts …) –- traces of residuals passed over from the distillation, impurities picked up from storage vessels, ingredients in the local water used, etc. But in the case of vodka (as opposed to more sophisticated boozes like whiskey and brandy) these additives are for all practical purposes very close to being nonexistent. What we are dealing with here is just a simple chemical preparation: a 40 % solution of ethanol in water. Period.
The Vodka Belt -- ethnic vodka-mixing
So the question of what mixes well with vodka should essentially boil down to clear-cut chemical considerations. But interestingly, there seems to be an ethnical side to vodka-mixing too. The name is derived from Russian, so we may assume that vodka mixes well with Russians. In fact, it seems to mix rather well with a number of other peoples living in the colder northern regions of Europe as well -– Finns, Swedes, Norwegians, Estonians, Latvians, Lithuanians, Poles – people living in the so-called “vodka belt” of Northern Europe. Here people are not enjoying alcoholic beverages for their exquisite taste and mild intoxicating effects. No, what these people are after is getting dead drunk as fast as possible, completely stoned in full Russian style, particularly on a Friday and/or Saturday night. So for the originators of vodka the stuff was never meant to be a tasty beverage to be mixed in sophisticated ways for culinary enjoyment. Rather, it was (and is) just an unpretentious chemical agent for getting pissed and paralytic as efficiently and cheaply as possible, like greased lightning.
Sugar and salt
Ethanol is a hydrophilic organic solvent, i.e. it mixes readily with water and various water solutions. Hence vodka may be diluted with whatever watery beverage you desire to use, in whatever proportions. Sugar has limited solubility in ethanol, but the 60 % water in vodka helps to dissolve appreciable amounts. If too much sugar is added, then un-dissolved crystals will remain. Skilled manipulation of the solubility properties at different temperatures of 40 % ethanol may result in the formation of large, nice-looking sugar crystals in the bottle.
If small quantities of some saline solutions are added to vodka, then precipitates of the salt may form, because many inorganic salts have limited solubility in ethanol. I have no idea why anybody would want to add small amounts of strong salt solutions to vodka, but now you are at least warned. (Well, come to think of it, I have no idea why anybody would like to mix vodka with anything, but this is of course beyond the scope of this writeup.)
The chemical secret of Pernod
On the other hand, the organic solvent properties of ethanol may dissolve organic oils and fats even in the presence of 60 % water. Small quantities of diluted water emulsions of oils (e.g. milk) may clear up when added to vodka. The solvent power of 40 % ethanol is actually used in the preparation of the French aperitif Pernod. Here Star anise oil is a chief ingredient in Pernod, forming a clear, yellow solution. But when water is added, then the ethanol becomes diluted to such a degree that it can no longer hold the oil in solution. Instead a whitish “precipitate”, i.e. an emulsion of small un-dissolved oil drops appears, making the water-Pernod mixture look like yellowish milk.
Idiosyncratic, but pedestrian
Taste is idiosyncratic, so mixing vodka, an essentially tasteless (pun intended) 40 % ethanol solution, with various other beverages may possibly elicit new sensations in the taste-buds. But as for me, a resident of the “vodka belt” of Northern Europe, I have a rather contemptuous attitude toward this completely pedestrian booze. I don’t feel that it mixes well with anything, least of all with people.