I know plain frozen yogurt exists, but what is so special about sugar that there doesn't seem to be a non-sweet substitute?
Sugar does more than just make ice cream sweet. It also controls the way ice cream freezes. Without it, you tend to get bigger ice crystals, which have an unpleasant mouth feel.
There are substitutes. Breyer's sugar-free ice cream, for example, has guar gum, polydextrose, cellulose gum and gel, and maltodextrin, among other things. Home ice cream makers don't have ready access to all of those ingredients, but there are recipes for making sugar-free ice cream at home. (I haven't tried them and can't vouch for them.)
As for making savory ice cream... yeah, people have tried. Most of those are nonetheless somewhat sweet. That may be in part because coldness really dampens flavors; people enjoy melted ice cream because it's even sweeter than cold ice cream. Umami flavors are hard enough to do well even warm, and they'll become very bland when frozen. Maybe if somebody wanted to really pack an ice cream with marmite and Parmesan cheese...
what is so special about sugar
Sugar is a pure short-chain carbohydrate. It is a solid which makes the bulk of the ice cream.
Ice cream has a narrow range of ratios of liquid, fat, and non-fat solids. If you throw out the sugar, you cannot replace it by something liquid, because the ice cream will have the wrong texture (not smooth enough). You cannot replace it with fat because then the ice cream will have the wrong texture, tasting like frozen butter, if you manage to churn it without breaking the emulsion at all.
You can in principle replace the sugar with some other carbohydrate. Long-chain carbohydrates have to be cooked, and if you put in enough to replace all the sugar, the whole thing will taste a bit too doughlike. Small amounts of starch are usable, and actually give you a nice texture because they bind water - then you get gelato. But you still need to "fill up" with sugar.
Short carbohydrates make decent substitutes for the sugar in ice cream. For example, you could use pure fructose - if you can stand the sweetness. The short-chain carbs we eat tend to be sweet, and calorie dense, so most people are not really interested in using them instead of sugar.
Some less common stuff could be used - sorbitol, maybe inulin - but beside being sweet, these things have side effects on the digestive system.
In principle, you could try using protein instead of the sugar. You won't get a perfectly smooth ice cream. The protein will surely soak up some water, but it won't dissolve in it, so it will stay grainy, and the non-bound water will stay around to produce crystals. Also, pure protein doesn't taste too good. And it is not that easy to source as a home cook, and somewhat too expensive for industrially produced food.
The most promising thing you could try would be binding the liquid with emulsifiers or other binders. Pectins are probably a good choice, then you will get something sherbet like. Then you will need something else to give flavor to the ice cream. The texture will of course not be the same as standard ice cream, but you might like it.
If you have access to exotic equipment, you can try whipping anything frozen into a mousse, and call it an ice cream. I think there was a cooking show which has made this into an injoke, but have not seen it personally. In any case, you'd need something capable of reducing solid frozen blocks of food to puree, without heating it enough that it melts. It is not really something that will become a widespread recipe.
The question was about unsweetened ice cream. Yet, you are all mentioning how to make it sweet using other carbs, fake sweeteners, or natural sugars. The point is for it to be unsweetened. And yes, fruit and honey have sugar in them...but, I digress.
I make unsweetened ice cream regularly. It can not be stored in the freezer, however. You have to make it in small portions and eat what you make.
I have an ice cream maker.
I pour 1 cup of cream into the ice cream maker.
Once the cream starts to get a bit thicker I add 1 to 3 of the following choices into the machine, depending on my mood. 1 tspn ginger, 1 tspn cocoa powder, 1 tsp cinammon, 1 tsp peanut butter powder, 1 tsp of vanilla, 1 tsp sugar free extract, 1 tsp flavored vodka (0 carb, 0 sugar), or any combination thereof.
Wait 20 minutes and it's done.
I enjoy putting a teaspoon of unsweetened peanut butter on top.
There is a difference between not sweet, and no added sugar etc. I make a lot of icecream at home, but I rarely use sugar. I mostly make some kind of yoghurt ice cream, and I use fruit for flavor. Which brings me to: Icecream can be sweet without sugar Most fruits have a sweet taste, and if you use them in icecream it will be sweeter even without sugar. Honey is also an option, I tried it a few times and it works really well.
But that depends on what exactly sugar added sweeteners is in your opinion. Do you mean the 'no carbs' way of sweet? Because then you're basically stuck with just frozen yoghurt anyway.
As for the problem with the ice crystals someone mentioned earlier: I foud that alcohol works really well. I once made Baileys icecream (without other sugar added) and that turned out really well. After that I added vodka to watermelon sorbet (also no sugar) and that worked way better than expected. And vodka is ofcourse almost without carbs etc.
So the big question for you really is: What exactly do you mean by sugar? The carbs, or just anything that is sweet?
In India now we have Paan flavoured Ice Cream, (https://en.wikipedia.org/wiki/Paan) which has both sweet and savoury flavours.
Ice cream, itself, is defined as "a sweet flavored frozen food containing cream or butterfat and usually eggs." Considering that, a dish that was not sweetened in some manner would, technically, not be ice cream. Rather, it would be a frozen, milk- or cream-based dish, which had similarities to ice cream.
However, there are adaptations that people use, which may not technically be ice cream, but are termed as such. Some examples could be found by searching for savory or unsweetened ice cream recipes. Many of these, however, would use fruit or other foodstuffs as a sweetener. With that being said, I have seen references to cocoa-based "ice cream", which would likely attract dark chocolate lovers like myself.
Just like chocolate mousse made with heated double cream and 85% coco chocolate like Green & Blacks or Lindt which has a low sugar content, you can make ice cream with those same ingredients but just not heat the cream. You have to keep stirring it every 2 hours and put it back into the freezer so that the texture is just right.