You could - provided you have a specific custom freezer that can reach as low as -17° F / -27°C - produce whiskey ice cubes.

The question is, should you?

If you are cooling your drink, you are significantly reducing the amount of volatile compounds that you can smell and taste. A purist would probably scoff at the suggestion to drop an ice cube or two into their prized single malt and the Cognac aficionado would lean back, gently cradling their sifter in the warm hand and shaking their head. But seriously, you would be dulling the taste of your cocktail and likely touching said whiskey ice cubes wouldn't be nice either.

Many cocktails are shaken with ice, this isn't to mix it - you could do that with a spoon - it's to dilute the drink with water. This seems counterintuitive, but that's usually desired. The idea of having a martini (essentially gin with some value-adds) without some dilution isn't appealing, they are already very strong. A typical martini is about 28% alcohol after shaking or stirring, using alcohol cubes would bring that up to almost 40%, which is a 42% increase in alcohol!

Cask strength whiskey is between 60-65% alcohol, water is added in the bottling process to bring it down to 40% typically, this is partly due to laws but mostly because over 40% the flavor is too strong. In fact, some single malt aficionados recommend adding a small amount of water to some whiskeys to 'open them up'.

So, alcohol cubes aren't widely used because you don't want them.

Freezing a mixture of two liquids solid is quite a complex process.

Contrary to most of the other answers, whiskey/vodka/gin/whatever 40% hard drink starts to freeze at ~ -30°C, but it is only the water freezing at first. It then proceeds towards lower temperatures to deplete the mixture of water until you have almost pure ethanol. This happens at ~ -110°C where the ethanol freezes as well.

(This is also how some wines are concentrated - the wine is frozen, the ice is removed. Ice takes away some acid as well, so the wine is generally improved. The whole thing is done at -20°C or -25°C and not lower.)

Thawing will proceed in the reverse order, releasing pure ethanol first and water next.

Believe me, a lot of things will go wrong.

0. Cryo gear working at -120°C is not cheap. What's worse, it is not trivial to work with.
1. You will need to stockpile different ice cubes separately.
2. A lot of substances will precipitate from the drink in the process. A lot of it will not dissolve afterwards. Instead of whiskey the ice will convert into a discolored liquid with solid residue.
3. The whole thing will become undrinkably cold, undrinkably strong and even dangerous to touch. Food/drink and health regulators will not be happy.

This is one of the rare "why not" questions which happen to have a technical reason behind them, rather than a random "it could have been done, but nobody cared enough to do it".

1. It would be way too cold. Hard alcohol seems to freeze somewhere between -25 and -30 C (I found different numbers when trying to look it up). You would need to get it even colder so it doesn't melt right away - these are "ultra low temperature" freezers, which go between -50 and -86 unless you want the fancy stuff (but luckily for you, alcohol has no DNA you need to preserve). Altogether, it would cool down your drink to the wrong temperature.

Here, we also have the point that Stephie mentions - there are also people who think that nobody should cool their drink even with 0°C ice cubes, and frequently the people who say that are the ones who are willing to invest the most in their drinking experience. But there are lots of people who like their drinks cooled with normal ice, and I think that even these people would prefer not to do it with ultra-cold alcoholic-drink-ice.

2. The logistics of it would be a nightmare. Getting dry ice and the required paraphernalia is not all an everyday task, and once you have it, you have to handle the stuff. It isn't impossible, but it is way more trouble than most people tolerate when they want a drink.

And if you would use an ULT freezer instead (ultra low temperature) - well, there are all kinds of issues around that too. There's the cost (lab equipment range!), the electricity consumption - and when you have it set up, you can't just go around throwing the door open every five minutes and sticking an ice cube tray of room temperature liquid into it, or grabbing one to get out (hopefully not with your bare hands!). A bar could probably build some kind of setup with blast chillers and a temp -18 storage for the daily batch of cubes, but that's an awfully expensive investment for a novelty gimmick, needs expertise and motivation more commonly found in engineers than in bar owners, and a financial gamble without a clear payoff.

In the end, it is expensive, complicated, and without real advantages.

No need for ice cubes

If you happen to store the (~40% ABV) drink in the freezer it won't freeze but will stay cold, I personally have one of the cheaper whiskeys in my freezer that I use for mixing. If you do freeze it that may not be comfortable for humans as it will hit -27 °C (-16 °F) or less.

Why people don't have whiskey cubes

1. Altered taste - As described by Stephie the low temperature alters the smell and thus taste of the drink, I think approximately to the same extent as the stones cooling the drink down.
2. Choice - People who drink whiskey for fun tend to have a variety of whiskey, in order to have the whole assortment in the freezer they would need to label and store all kinds of whiskey separately.
3. Taste change - Air humidity condenses on all surfaces inside the freezer, including ice cubes, it dilutes the drink just as much as if it were a metal, stone or porcelain cube. In addition if there is any smell in the fridge that penetrates frozen stuff it will stick to the expensive whiskey cubes as well. The same could happen to metal or stone coolers but one can simply wash the smell off them.
4. Freezer temp - The difference between recommended freezer temperature of 0 °F (-18 °C) and the freezing point of alcohol increases the operating costs of the freezer, may cause freezer burn of other items and increase the amount of frost in the freezer.
5. Taste of other frozen items and general quality of life - given that whiskey has a lower melting point than likely any other item in the freezer it will thaw first. This means that one would need special containers in order to prevent it from seeping out and/or evaporating and ruining frozen food.

What you can do instead of having whiskey ice cubes

1. Store stones/metal/porcelain cubes in the freezer - this is a general good advice, no real downsides as long as you let guests decide whether they want them or not, make sure you store the stones in their boxes to reduce frost
2. Store the whiskey in the freezer - this reduces visibility of whiskey, alters the taste, minimal frost
3. Store glasses in the freezer - not convenient but works, make sure you store the glasses in at least a plastic bag to reduce frost