OK, I gotta wade into this morass . . .
Temperature very simply is a measure of hot and cold. Thermal energy flows from hot to cold, so temperature is a convenient way to track this. If you get much beyond this, it isn't simple any more.
The precise relation between temperature and energy implied from the Boltzmann constant applies to ideal gasses. Among other things, ideal gas molecules do not take up physical space and do not interact at a distance. At low pressures and high temperatures, many gasses act very like ideal gasses.
Back to the Arctic. At low temperatures, all gasses act less and less 'ideal'. And water vapor in particular (with strong intermolecular forces) is a particularly non-ideal gas.
The relation between temperature and energy in an engineering sense (with real materials) is defined as heat capacity. In an ideal gas, this is a constant, in real materials, it varies with temperature. Ice (and water vapor) has a heat capacity very roughly one-half that of liquid water. Changing one unit mass of water by one unit of temperature will change the temperature of two unit mass of ice or water vapor by the same magnitude of temperature, or twice the temperature difference in the same mass.
When matter freezes or vaporizes, energy is involved in the state change beyond that which changes temperature. In the case of water, these changes (at a constant pressure) happen at a particular temperature. Ice and water in contact (at a given temperature) will always be at the same temperature (the melting point) no matter what the proportion of ice and water. Adding energy increases the proportion of water until the ice is gone, at that point the temperature again begins to rise as thermal energy is added.
The heat capacity of ice (real ice, in real conditions) is not constant. Very roughly, ice at -40 degrees has only 90% of the heat capacity of ice at the freezing point, and heat capacity continues to fall with temperature from there. The actual thermal energy in ice is the heat capacity integrated over temperature (the area under a heat capacity graph) from absolute zero to the temperature of interest. Absolute zero is the point at which there is no thermal energy present. This is in only in a rough practical sense - it turns out that (due to quantum effects and other weirdness) things are more complicated than that, but not in a way that affect understanding of the Arctic.
As a practical matter, it is almost always the change in thermal energy (or relative amount) we care about - this is why absolute temperature scales are rarely used in everyday life. Common uses for them include calculating the efficiency (among other things) of heat engine cycles, and describing temperatures so low that the freezing point of water is no longer a meaningful reference. They are also handy for calculating heat transfer by radiation to free space (which behaves much as though it were at absolute zero.
Generally speaking, conduction and convection of thermal energy are proportional to temperature difference. This is why 'degree days' are so handy. It is critical when computing temperature differences that the same scale is used - mixing relative and absolute scales or scales with different increments (F and C, K and R) will lead to wrong answers.
So to go back to the original question from Jim Williams, Matt Strassler's explanation was not particularly helpful. While it is true temperature is a measure of mechanical energy per molecule (sort of), there are lots of modes by which a molecule can hold energy. In a (monatomic) ideal gas, it simplifies to translational motion, which is great for classroom physics. In solids, liquids, and real gasses (particularly water vapor) it is much more involved, and changes with temperature. It is not true (nor particularly significant) that "-172.15°C is 1% more energy than -173.15°C". At 100K water is ice, and its heat capacity is about two thirds that at the freezing point and dropping somewhat faster than linearly with temperature. A quick look did not turn up properties of water ice at lower temperatures, but very roughly, it is reasonable to expect the total thermal energy of ice at -173.15C to be a bit less than 2% higher than at -172.15C.
A later question was "At what temperature is there a 1% energy increase from 0oF [~-17.77oC]?". Again, I can only extrapolate (and also establish reasonable bounds on) properties below 100K, but the answer (by integrating heat capacity with respect to temperature) looks to be very roughly -16.1C or 3F (in other words, from 0F/-17.77C, a 1% increase in total thermal energy will yield a temperature increase of about 5/3rdsC/K or 3F/R). This answer is significantly lower than you'd get by assuming constant heat capacity.
It is fair to say that to speak of a 1% (or any other proportion) change of temperature in a absolute sense on any temperature scale that is not absolute (e.g. F, C) is completely meaningless. It is also, unfortunately, true that for all but a few specific thermodynamic calculations that to speak of a particular proportional change in temperature (in an absolute sense) is also often all but meaningless. Working with proportional changes in temperature difference, over which there is no phase change and the overall span of temperature is reasonable, is perfectly fine. It is ok for you to feel twice as cold, it is without physical meaning to say any particular temperature is twice as cold as some other temperature.
In the words of NIST (
http://cryogenics.nist.gov/Papers/Cryo_Materials.pdf),
Models for specific heat began in the 1871 with Boltzmann and were further refined by Einstein and Debye in the early part of the 20th century. While there are many variations of these first models, they generally only provide accurate results for materials with perfect crystal lattice structures. The specific heat of many of the engineering materials of interest here is not described well by these simple models.
I'm not sure this makes things clearer (or will make Jim Williams or others feel better) but the fact is thermodynamics is not a simple science, and many of the simple assertions made above do not stand up to scrutiny.
The good news is, Arctic temperatures may be cold but they are not truly cryogenic. Properties of ice (and water vapor) at Arctic temperatures are well described in engineering literature (and, presumably, have been correctly applied in models heavily dependent on thermodynamics, such as PIOMAS). At some point you have to trust the experts or do a LOT of homework.
Please don't even attempt to understand the physics (or, too often, complete lack of same) behind wind chill numbers. And please don't try to apply wind chill values when calculating degree days or computing your heating bill . . .