Summary: Most types of sugar will absorb some heat as they dissolve. Commercial baked goods often use a type of "snow sugar," which is designed to be used on warm, moist foods without dissolving. It's likely that commercial doughnuts use a lot of dextrose in their "snow sugar," which requires four times as much heat to dissolve as the sucrose in normal powdered sugar, leading to a "cold" sensation when consumed.
I tried poking around a bit in various sources and can't find a definitive answer, and I personally can't recall experiencing this "cold" sensation. But it seems to me the explanations really can only come down to two possibilities:
(1) The sensation is due to ingredients that stimulate nerves which feel "like cold." The standard example for this is stuff like mint or eucalyptus, whose menthol and eucalyptol create a sensation of coolness even without drawing heat. The problem with this explanation is that the list of chemicals that cause this sensation is limited, and I don't know why they'd include any of them in powdered sugar. That said, snow sugar products, as mentioned in comments, can contain various organic compounds to keep them from melting/running on warm moist baked goods. I suppose it's possible -- though unlikely -- that some doughnut coverings contain something that would do this.
(2) The sensation is actual cold, i.e., the increased heat transfer from tongue to donut, resulting in a perceived temperature drop. That would be due to "thermodynamic mumbo-jumbo" mentioned in the question, i.e., heat/enthalpy of solution/dissolution. That's just a fancy term to say that it takes some energy to separate a crystalline solid (where molecules are bonded together in an organized lattice) into individually dissolved molecules in a solution.
The problem with the second explanation is that we consume food all the time and don't generally sense the quite tiny amount of heat transfer it takes to dissolve things. But powdered sugars are somewhat unusual foods in that (1) they exist as a crystalline solid with a high solubility, and (2) they are ingested with a fine particle size that increases surface area significantly and therefore rate of dissolution.
It's likely that the mixture of chemical components used in some commercial sugar mixtures like snow sugar are engineered to have an increased enthalpy of solution (and thus don't appear to "melt"/dissolve or run on warm baked goods). While a warm baked good may not have enough moisture and heat to dissolve them, our mouths do, and that increased heat necessary for dissolution is perceived as "cold."
What exactly could the specific chemical be? Some comments have suggested dextrose (which indeed is the primary ingredient in the King Arthur version of snow sugar as well as in the French product Sucraneige). Dextrose (the D-isomer of glucose, and its most common form) has a endothermic heat of solution that's about double that of sucrose. That means that "snow sugar" needs to absorb about four times as much heat per weight as normal sucrose-based confectioners sugar when dissolving. (I'm assuming that the other components of the snow sugar are relatively minor; they'll have significantly worse solubility.)
Particle size, as mentioned, is also an issue. The Sucraneige mentioned above claims 10X fineness, but many powdered sugars at the grocery store will also be the same. If you are comparing a 10X sugar to a coarser powdered sugar, that will also make a difference: smaller particles have more surface area and can dissolve faster. If it dissolves faster, the reaction happens faster, thus absorbing heat more quickly (and feeling colder).
Still, this amount of heat transfer seems quite small. But I suppose for those who notice it, it could explain the difference. A quick search of some common commercial powdered sugar doughnut brands (like Hostess and Entenmann's) doesn't clarify the powder ingredients, but dextrose is listed quite high in the ingredients list, just after flour in one of them, seeming to hint at some form of "snow sugar." Other non-powdered types of doughnuts seem to have dextrose listed far later (if at all).
Technical note: For those who are wondering, why four times the enthalpy? The enthalpies that are given in the linked table are 5.4 kJ/mol for sucrose and 11 kJ/mol for glucose/dextrose. But these are per mole, and the molecular weight of sucrose is roughly twice that of glucose. Therefore, to get the same amount of glucose by weight, you'd need roughly twice as many moles of glucose. Since glucose already has twice the enthalpy of solution, the same weight of glucose will have roughly four times the enthalpy of the equivalent weight of sucrose. Also, it should be noted that perceived sweetness of dextrose/glucose is somewhat less than sucrose (roughly 70-75% as sweet), so more of it tends to be used. I don't know that that's a major factor here though, where doughnuts are doused in lots of sugar either way.