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".
- 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.
- 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.